Elementary Theory of L-functions and Eisenstein Series (London Mathematical Society Student Texts 26)

8

1: Algebraic number theory

the formal real span of a and thus FR = Rd as real vector spaces. That is, FR/# = (R/Z)d, which is compact. For any point x e F, we can find a small neighborhood U in FR SO that U is isomorphically projected into FR/O. Thus, for any fractional ideal 6 containing a, the image of B in FR/# is discrete because B has only finitely many generators over Z. Since FR/# is compact, 61a is a finite group. We now define for all fractional ideals a

Two ideals a and 6 are said to be relatively prime (or a is prime to F), if a+6 = O. We then have an exact sequence 0 - > cC\B -> O -> (O/a)®(O/6) - > 0 x h-> (x mod a)0(x mod 6). Thus N(cC\S) = N(a)N(B) if a+£ = 0. Since Oz> a, multiplying by B, we see that 6= BO ID aB. Similarly we see that az> aB and hence cf]Bz> aB. If a+B= 0, we can find a e a and b € £ such that a+b = 1. Then we see that x = xa+xb for any x e flfl£. Thus x e aB. This shows that cC\B = aB if a+6 = O. That is, we have (2a)

N(a£) = N(a)N(S) if a+£ = a

Let L and L' be free Z-modules of finite rank. Let ( , ) be an inner product on LxL' which is non-degenerate on both sides. Thus rankz(L) = rankz(L'). Let ei be the smallest positive integer in { ( x , y ) | x e L and y e L ' } . We take xx and yi such that (xi,yi) = ei. Let Li = {w e L | (w,yi) = 0}. Then L = xiZ©Li. By definition, xiZflLi is orthogonal to everything and hence is reduced to {0}. Let Z E L with (z,yi)^0. By the division algorithm, (z,yi) = eiq+r with 0 < r < e i . Then (z-qxi,yi) =r. By the minimality, r has to be 0, that is, z-qxi G Li, or in other words, xiZ+Li = L. We also know that (L,yx) c eiZ. Similarly L' = yiZ©Li' for Li' = { w e Lf | (xi,w) = 0}. Repeating this process, we find two bases xi, ..., xa of L and yi, ..., yd^ L' such that ei = (xi,yj)8ij for the Kronecker symbol 8ij and ei I ei+i. We can apply this argument to the inner product (x,y) = xAly for x,ye Z d and a matrix A e Md(Z) with det(A)^0. Then we see I det(A) | = det((xi,yj)) = Ef=1ei = #(Z d /Z d A). Applying this argument to A = p(a) for a e O, we know that (2b)

iV(aO)= I//(a) |.

Thus N: / - ^ Q x coincides with the norm map on ^P up to absolute value. Thus N is a homomorphism of multiplicative group T. Now take an ideal a

1.2. algebraic number fields

9

and its basis a = (ai,...,0Cd). We identify a with Z d via this basis. We then define the regular representation p f : F -> Md(Q) by aoc = (accOi = X j = i a Ji a J

= a

P'( a )

with

P'
If we write a = coA for a basis co = (coi,...,O)d) of 0, we see Ap'(a)A"1 = p(a). Thus N(a) = det(p'(a)). Then applying the above argument replacing O by a, we see that I N(a) I = #(fl/aa) = N(a0). (2c) Finally from the equality #(O/aa) = #(O/a)#(fl/aa), we conclude that (2d)

N(?La)=N(aO)N(a).

Combining (2a-d), we see that (3a)

N: / -> Q+ = {a e Q | a > 0} is a homomorphism of multiplicative groups.

Indeed, for any integral ideals a and 6, we can find a e 0 such that aa+£ = 0. Then we see that and hence (3a). Since A^K/Q : /—» Q+ for any finite extension K/F is a multiplicative map satisfying NK/Q^OK) =A^F/Q(aO)[K:F-5 for a e F x , we see easily that (3b) for fractional ideals a of F, To prove Theorem 1, we first prove the following lemma: Lemma 2. There exists a constant C depending only on F such that any ideal a in O contains a non-zero element a e a with \N(a) I < CN(d). Proof. Let O- Zcoi+---+ZcOd. Let g be the greatest integer [^JN(a)] not exceeding ^]N(a). Consider the set of numbers S = {niCQi+—+ndcode o\ 0 < n i < Then #(S) = (g+l) d > N(a) = #(O/a). Thus there are at least two distinct elements P, y G S such that p = y mod a. That is 0 * a = p-y = mi©i+- • •+mdC0d with I mi | < %lN(a). Let M = maxij( | coi(j) |). Then |N(a) | = Yl*=l

| micoi(j)+.• •+mdcod(j) I < ddMdA^(a).

We can take d d M d as C. We now give the proof of Theorem 1 (which is due to Hurwitz). Let c be a class in II(P. Take an integral ideal a in c"1 and choose a as in the

10

1: Algebraic number theory

lemma. Since a => ocO, we can find another integral ideal 6 such that aB = ocO. Thus Be c. On the other hand, N(a)N(B)= \N(a)\
Minkowski's estimate: C < UD^\

t d*

(see below);

- I | Dp I when F is totally real [Si]; here Dp = det(Tr(o>iCOj)) for a basis {coi,..., C0d} of O is the discriminant of F/Q, t is the number of complex places of F and r = -

or - +1 according

as d = 1 mod 6 or not. (Here [a] is the greatest integer not exceeding a.) Exercise 1. Let &+= {ao\

oce F, oc a >0 for all field embeddings a of F into R},

I(m) = {5 = - I n and d are integral and prime to m) for a given ideal m, and H?+(m) = &-f){aO \ a e F x , a = 1 modxm}, where a = 1 modx7rc means that aO e I(m) and there exists |3 e fP+ such that pOe /(/n), (3G O, a p e O and a p ^ p m o d w . Then show that &+(m) is a subgroup of finite index in I(m). The finite group CLp(/n) = I(m)/^P+(m) is called the strict ray class group modulo nu Exercise 2. (a) Using (4), show that the class number h(F) = 1 for the following fields: Q(Q, CKC+C'1) for C7 = 1 but £ * 1, and Q(3V2). (b) Show that h(F) > 2 for F = Q(V:5). Exercise 3 (Minkowski). Using the estimate (4), show that | Dp | > 1 for any number field F & Q. An ideal p of O is called a prime ideal, if the residue ring O/p is an integral domain (i.e. having no zero divisor). When a prime ideal p is non-zero, O/p is

1.2. algebraic number fields

11

a finite integral domain. Thus for every 0 * a e O/p, the sequence of elements a, a2, a 3 ,..., a11,... in O/p has to overlap. Thus for some i > j , a ^ a ^ . So Thus we can always find a positive integer h such that a h = 1. ai-j _ i Therefore a"1 = ah-1 G O/p. Thus every non-zero element of O/p has an inverse and hence O/p is a finite field. Since ideals of a field are either {0} or the total ring, there are no proper ideals in O containing p. Thus p is maximal. By definition, the ring O is normal (i.e. any element in F integral over O belongs to O). A normal integral domain whose non-zero prime ideals are maximal is called a Dedekind domain (for further study of such rings, see [Bourl, VII.2]). For any ideal a, by Zorn's lemma, there exists a maximal ideal p containing a. Then p'la is again an ideal of O. In fact, multiplying pz>a by p"1, we get O = pp~l 3 ap~l. Thus a=pB with an ideal B of O, and we have N($) 0 for all p. Proof. The ring O/p has no zero-divisors, for if a,b G O and ab G p, then either a G p or b G p. We now show for ideals a and B of O, if a prime ideal p contains aS, then either pz> a or p z> B. We may suppose that a is not contained in p. Thus we have a G a with a e p. Since for each b G £, pi) aS & ab, we see b G p and hence pz> & We now suppose that an ideal a in O has two prime decompositions a = p-^p2- • *pm = qx # 2 "' #n l o w i n g repetition of primes. Since either p a o r p ^ if p 3 a£, by renumbering the prime ideal ^ ' s , we may assume that p1 3 qv Since q^ is a prime ideal, it is a maximal ideal, and hence px = qv Dividing both sides by pv we get a new identity Vi 'Tm = #2*"&i* Repeating the above process, we finally know that m = n and pi = ^ for i = 1, ..., n after renumbering the ^ ' s . This shows the uniqueness of the prime decomposition. By definition, each fractional ideal / can be written as a f- - for integral ideals a and S. By the uniqueness of prime factorization for b

a and fB and 6, the ideal / has a prime factorization which is unique. If p and q are distinct maximal ideals of O, then p+q = O. Taking p G p and q G q such that p+q = 1, for any given positive integer n, we can find a large N such that 1 = (p+q)N e pn+^1. Thus pn+
12

1: Algebraic number theory

prime factors in both the denominator and the numerator. We write (a,S) = 1 when a and S have no common prime factors. Let ja(F) be the group of all roots of unity in O. Let Oi,..., a r : F -» R denote all distinct real embeddings and choose complex embeddings Gr+i,..., G r+2t : F -> C so that {o r+ i, ..., Gr+t,ar+t+i = co r + b ..., a r+2 t = ca r+t } makes the set of all complex embeddings of F into C, where c denotes complex conjugation. We simply write a® for a?1 for a e F. For each fractional ideal a of F, taking a basis a = (ai,..., ad) of a over Z, we find the following formula: (5)

I det(ccj(i)) | =

for D = D F .

To see this, we embed F into FR = R r xC l by i : a n (aw)i=i,...>r+t. Then we define a measure dji on FR by ®ir=1dxi®t=11 dxjOdxj |, where for a variable z = x+iy on C (x, y e R), |dz®dz| = 2dxdy. Then for d(=r+2t) linearly independent vectors VJ = (xij)i==i>...>r+t e FR, it is an easy computation to see that the volume with respect to d|i of the parallelotope V = V(vi, ..., Vd) spanned by vi, ...,Vd is given by x

1,1 x

x

l,2

l,d

r+t,l

x

r+t,2

x

T+1,1

x

r+l,2

x

•r+t,l

x

r+t,2

x

r+t,d r+l,d

r+t,d

Thus writing u

i a (2)

A = A(a) = A(a) =

! ,al

we see that

a21}

-

a^

^(2) U 2

n(2)

a2

fy (d)

^d

ad j

| det(A) | is the volume of V(i(ai),...,i((Xd)), which is the funda-

r. mental domain for F R / # . Thus we see N(a) = [CKa] = |det(A(O))|

Since

A l A = (Tr(aiaj)), we get (5). Now we want to determine the structure of the unit group (?. Let |Li(F) be the group of all roots of unity in O. We simply write oc^ for a*1 for a e R First we prove

1.2. algebraic number fields

13

Lemma 3 (Kronecker). For an integer a <= 0, if \ a then a is a root of unity.

= 1 for all i = 1,..., r+t,

Proof. We can identify F ® Q R with R r xC l so that the projection of F to the i-th factor R or C is given by Oi (see (1.5b)). Since a basis of O over Z is a basis of F over Q and hence is a basis of F ® Q R over R, O is a (closed) discrete Z-submodule of R r xC l (because identifying R r xC l with R d via the basis of O over Z, we know that the image of O in R is Z ). On the other hand, S = {(xi)e R r xC l | |xi| = l } = {±l}rxSit (with the circle Si of radius 1 in C) is a compact set. Thus CHS is a discrete and compact set and hence is finite. Let |1 = {XG O\ | x ^ | =1}. Then \i is a subgroup of O* and is contained in SflO and hence is a finite group. Thus ja is made of roots of unity. Lemma 4 (Minkowski). Forgiven

n-real linear forms Li(x) = S =1 ai q x q on

n

R , suppose that D = det(apq) ^ 0. Then for any positive numbers Ki,..., Kn such that II^Ki > ID |, there exist

m = (mi, ..., m n ) e Z n with not all mi

zero such that \ Li(m) | < Ki for all i. Proof. Consider the parallelotope PQ = (x e Rn | |Li(x)|
for all i}.

Similarly we consider the translation P m = {x G Rn | | Li(x-m) | <j for all i} whose center is an integer vector m. We can give the following illustration of the situation when n = 2: If P m and P m . for m ^ m 1 have non-trivial intersection, then for x G PJiPm 1 , we have m)| < y

and

|Li(x-m')| <-y for all i. f T S o w e s e e Lx(x) = -1 Lj(x) = 0 I Li(m-m') I < Ki. Thus what we need to prove is that if IIf=1Ki > ID |, then P m and Pm- intersect for some m ^ m1. We shall show that if Pri/lPm' = 0 Ilf=1Ki < | D | . In fact then the volume S(L)={XG

n

R | |xj|
for any given

c = Maxxep0{ | xi |,..., | xn | }. Then for

for all m ^ m1, then

J

of the

L

is less than n

P m 's in a square 2nLn.

Let

m e Z , | rrii | < L if and only if

14

1: Algebraic number theory

S(L+c)z>Pm. There are (2L+l) n such m's for each integer L. This shows that (2L+l)nvol(P0) < 2n(L+c)n. By making L large, we know that vol(Po) ^ 1 2n(L+c)n since lim — — = 1. On the other hand, we see that L^oo(2L+l) n

vol(Po) = J-'-Jp dx 1 dx 2 ---dx n ^

Kn.

This shows that Ki Kn < | D | . By our assumption, P m nP m ' = 0 for all m * m\ Po does not contain any non-zero integer point. Since Po is a compact Min (| xi-yi I,..., I xn-yn |) set, and Z n -{0} is a closed discrete set, 8 = xePo.yeZMO}

exists and is a positive number. Thus if 0 < e < 8, the system of inequalities |Li(x)|
F

such that for any given positive numbers

Ki, ..., Kd such

that

IIi=1Ki > CN(a) and Ki = Ki+t for r < i < r+t, we can find a non-zero element as

a such that | ocw | < Ki for all i.

Proof. When all the field embeddings O[ (i=l, ..., d) of F into C actually take F into R (i.e. F is totally real), then we apply the above lemma to linear forms Li(x) = Sj=10Cj^Xj for a basis cci,..., 0Cd of a over Z. Then (by (5)), I det((Xj(l)) | = N(d) | VD | for the discriminant D = D F of F and hence is non-zero. Let C = £+1 VD | for some e > 0. Then for any set of positive numbers Ki, ..., Kd with Ki Kd ^ CN(a), we can find slightly smaller K'i< Ki such that Ki Kd > CN(ri) > K'I Kfd > N(a)|VD|. Applying the lemma to K'i, we find O ^ m e Z d such that |Li(m)| < K'i < Ki. Then for 0 * a = m i0Ci+***+md0Cd s a, I a^ I r . Then the desired assertion is true again for C = £+|VD1 for any e > 0. Exercise 4. (a) Write down all the steps of the proof of the above corollary when F is not totally real. (b) By using the argument of the proof of Lemma 4 and Corollary 1, show that we can take |VlD | as the constant C in Lemma 2.

1.2. algebraic number fields

15

Now we prove the Dirichlet unit theorem: Theorem 3 (Dirichlet). There exists a free Z-submodule E of rank r+t-1 in 0* such that C? = ji(F)xE, where we consider the multiplicative group E as an additive Z-module. Proof. First we shall show that the multiplicative group cf contains r+t-1 multiplicatively independent elements (i.e., if one writes the group cf additively, we shall show that in b*, there are r+t-1 linearly independent elements). Fix a basis C0i,...,C0d of O over Z. By Lemma 4, we have a positive constant C so that we can find an integer vector m e Z d such that | S^mjCGi^ | < iq for all i if Ki Kd > C; thus, we can find a non-trivial algebraic integer O ^ a e O such that I a ^ I < Ki for all i. In particular, applying this to Ki = *Vc for all i, we can find 0 * a e 0 such that |N(a) | < C. Then we redefine Ki = | oc(l) | for i > 1 and Ki = CI a ( 1 ) | /1 N(a) | and applying Lemma 4, we can find 0 * cci e 0 such that | ai ( 1 ) | < C -i r and | ai ( l ) | < | a (l) | for all i > 1. In the same |iV(a)| manner, we can find inductively 0 ^ a n G O such that loc n (1) |
foralli>l.

Then I N(a } I - n d I nL.^ I < c ' a n A ! ' n d \ i y / \ S ^ n ) \ — AJi_i I tX-n

I

I A r/

I rv , a) I - r

|11^_2 I ^ n - 1

I — v—

Since the number of ideals in O of norm less than C is finite, there are integers 0 < p < q such that o c p 0 = a q 0 . Namely ei = aq/Op is a unit and 1 £i (i) I = 10Cq(i)/Op(i) I < 1 for all 1 < i < r+t. Since | N(e) \ = 1, we conclude lei^H > 1. Similarly, we can find a unit £j (j = 1, ..., r+t-1) such that I £j (l) | < 1 for all i * j and 1 < i < r+t and | £j^ I > 1. Now we want to show that £i,...,e r+t _i are multiplicatively independent. That is, we want to show that the relation (6) Ceini es118 = 1 for C e |H(F) and s = r+t-1 holds only when C, = 1 and ni = 112 = ••• = ns = 0. For a e F, we write /i(a) = log(I a ( i ) I) for l < i < r and h(a) = 2log(\a(i)\) for r < i < r + t . Then the multiplicative independence of £j is equivalent to

(7)

d e t ( R ) ^ 0 for R = / s (£ 2 ) ... /S(£S)J

16

1: Algebraic number theory | E J ( I ) | < 1 for all i * j

Since

and l < i < r + t and

l e / ^ | > 1, we know that

Zi(ej) < 0 if i * j and /j(ej) > 0 and

Generally det(aij) ^ 0 under the following three conditions: (i) aij < 0 for i ^ j , (ii) an > 0 for all i and (iii) ^

S

,aji > 0 for all i.

In fact, if xA = 0 for A = (aij) and a non-trivial (xi, ..., x s ) e R s , then we see that aipXi+---+aPpXp+---+aSpXs = 0 . However, from 5L 1 aj p >0, we know that a p p > -(aip+---+ap.ip+ap+ip+-"+aSp) and from a pp = | a pp | and | aip I =-ai p for i & p, we know that I appxp | > -(ai p +- • •+a p .ip+a p+ ip+- • -+asp) I x p | if x p * 0. Then taking the index p so that I x p | = max( | X[ | ) , I a p p x p | > | ai p +- • •+a p -i p +a p+ i p +- • -+a sp I I x p | ^ I ai p x p | +• • •+ | a p _i p x p | + | a p + i p x p | +• • •+1 a sp x p | (because a ip = -1 aip |) > | aipXi | +• • •+ | ap_ipXp_i I +1 a p + i p x p + i | +• • •+ | a sp x s | |

- • •+a p .i p x p .i+a p+ i p x p+ i+- • -+a sp x s |,

which is a contradiction. Thus A is invertible. We conclude that the only possible solution of (7) is ni = n2 = ••• = n s = 0 and hence £ = 1. Next we shall show t h a t H = {ei n i .---e s n s I n = (ni, ..., n s ) e Z s }

is of finite

index in cf, which finishes the proof of the theorem. We consider e = (/i(e), ..., / s (e)) e R s for e e 0*. Since R is invertible, we can find a solution x e R s such that Rx = e; that is 1 8 ( i ) | = | (ei (i) ) Xl However, this is true even for n

(e s (i) ) Xs I for all i < s-1.

i=s

because

| N(e) \ = |iV(ei)| = 1.

Let

i = t i] be the maximal integer not exceeding Xi and define r\ = ei .---*e s ns . x

Then

ni

| (en" 1 )^ I < Sup ((ei (i) ) Xl

(es(i))Xs) = M, which is independent of the

0<xi
choice of e. The number of integers with

| a ( l ) | < M for all i is finite and

hence (O*:H) < +oo. Exercise 5. Show that for any proper subset S ^ 0 exists a unit e e (? such that i e I-S.

of I = {1, 2, ..., r+t}, there

I £ (l) I < 1 for i e S and

I £ (l) I > 1 for any

1.3. p-adic numbers

17

§1.3. p-adic numbers We briefly recall the construction and properties of p-adic numbers in this section. We refer to [Kb] for more details. Recall the construction of real numbers out of rational numbers using Cauchy sequences. A sequence {an}nEN of rational numbers is called a Cauchy sequence if for any given small positive e, there exists a large integer M such that if m,n > M, then | am-an | < e. Then defining an equivalence relation on the totality C of Cauchy sequences by {an} ~ {bn} if I a n -b n I -> 0 as n -> °°, we define the set of real numbers R to be Cl~. By assigning the constant sequence {a} to each rational number a, the set of rational numbers Q is embedded into R. The multiplication (resp. addition) of two Cauchy sequences {an} and {bn} is defined to be the Cauchy sequence {anbn} (resp. {an+bn}). This well defined ring structure induces a field structure on R which makes R an extension field of Q. By construction, any Cauchy sequence of R has its limit inside R. Obviously we can perform the above process of completing Q for any absolute value II || which is a function on Q having values in the non-negative real line (and may be different from the usual absolute value | I) satisfying ||a|| = 0 <=>a = 0 and ||a+b|| < ||a| + | b | . We can extend the absolute value to the completion by

b = lim b n

for the

number b represented by the Cauchy sequence {bn}. We list some examples of different absolute values in a somewhat more general setting. Let F be a number field and O be its integer ring. We fix a prime ideal p * (0) of 0. For any O ^ x e F, we decompose the ideal xO into a product of (non-zero) prime ideals in O as guaranteed in Theorem 1.2.2. Let v(x) = vp(x) be the exponent of p in xO. When the prime ideal p does not show up in the prime decomposition of xO (i.e. xO is prime to p), we simply put v(x) = 0. We also put v(0)=oo (i.e. (0) is divisible by p infinitely many times). Then we put | x | p = N(p)'v^x\ Then x h-> | x | p is a function having values in the non-negative real line, | x | p = 0<=>x = 0 and | x | p = 1 <=> xO is prime to p. It is easy to check the following fact which implies the triangle inequality || a+b || < | a || +1| b ||: (1) | x+y I p < max( | x | p, | y | p), and the equality holds when | x | p *

\y\F

Let us now compute I n! | p for a given prime p and n e N. Let m be a unique integer such that p m < n < p m + 1 . We define a sequence of integers 0
18

1: Algebraic number theory

and define ai and n2 by ni = aipm"1+n2. Thus after having defined ai and ni+i, we define ai+i and nj+2 by n^^a^jp 111 " 1 " 1 +n i+2 . Then we have n = aop^aip" 1 "^—Ha m with 0 < &{ < p. Let s = ao+ai+---+am. Then we claim that vp(n!) =—r. Writing kj for the number of multiples of p1 in the integers from 1 to n, we see that lq is the largest integer not exceeding —, i.e., P and thus Thus we have (2)

| n ! | p = p(s"n)/(p-1) for n e N.

Here is another example. Let a : F —»C be a field embedding. Then we define | x | a = | x ° | using the usual absolute value | | on C. Then obviously | | a is an absolute value. We denote by Z the set of all absolute values on F. Two absolute values are said to be equivalent if they give an equivalent topology on F. Here the topology associated to a norm | | is given by the metric p(x,y) = I x-y |. Each equivalence class of absolute values of F is called a place. Thus we have attached a place both to each maximal ideal p and to each complex embedding a. It is known that the set of places of F corresponds bijectively to the disjoint union of the set of all (non-zero) prime ideals of O and the set of all complex embeddings of F modulo right multiplication by complex conjugation (see [Bourl, VI.6.4] or [Wl, I]). Anyway, we hereafter identify the set of all places of F with the disjoint union of the set of all (non-zero) prime ideals of O and the set of all complex embeddings of F modulo right multiplication by complex conjugation c. Exercise 1. Show that I I a and | | T are different as places of F if the two embeddings a and x into R are different. Let I I v be a place of F. Then v is either a complex embedding or a prime ideal of O. We perform the completion process (sketched for Q for the usual absolute value) for this absolute value | | v- The resulting field will be denoted by F v . Thus F v can be identified with the equivalence classes of Cauchy sequences under | | v , and there is a natural embedding of F into F v . When v = p, Fp is called the /?-adic completion of F. When F = Q, any (non-zero) prime ideal of Z is spanned by a unique positive prime number p. In this case,

1.3. p-adic numbers

19

we write Q p for Q(p). The closure of Z in Q p is called the p-adic integer ring, denoted by Z p . Let n be an integer. Then we can find a unique integer [n] p such that n = [n]p mod p and 0 < [n]p < p. Let ao = [n] p , and applying this process to n p

—^k, we define ai as *

. We iterate this process and define, after Jp

L

having defined am, am+1 = r ^

Jp

{bm = ao+aip+a2p2+---+ampm}mez is a Cauchy sequence converging to n under | | p , and we can write n =

Sm=o a m p m • We can inductively solve the formal equation {Z°° rtampm}{Z°° xmpm} = 1 if ao is prime to p (i.e. n is prime to p). m=0

m=0

By definition, every p-adic integer z in Z p actually has such an expansion z = Z°° an(z)pn for unique integers an(z) with 0 < an(z) < p. In particular, z is n=0

invertible in Z p if and only if ao is prime to p. Thus all the ideals of Z p are exhausted by pnZp and they are all distinct. We can write n p Z p = {x e Q p | | x | p < p" n }. Thus Z p is a valuation ring and hence is a principal ideal domain. In particular, all finitely generated torsion-free modules over Z p have a basis (i.e. are in fact free). Let Op be the closure of O in F^ Taking an element G5 in O with v^(G5) = 1 and fixing a representative set R in O for O/p, we can expand in the same manner as in the case of Z p any ze Op into a unique expansion

z=

lZ=o^z)mn

with an(z)G R

-

Thus again all the ideals of Op are of the form U5nOp for some n (allowing n = oo), and therefore Op is a valuation ring. The above process of expanding ze Op into a power series of G5 can be performed in the finite ring O///11 and of course in this case the series is a finite sum: z m = 5^~oan(zm)G3n for Zm = z mod y11 with an(zm) = an(z) mod p m . Then the sequence {zm e O/f^}m satisfies z m = Zn mod p n whenever m > n. Thus z naturally determines an element z=limz m in lim(O/pm). m

m

On the other hand, for any given element

z. =limz m , we can recover m

z = ^ n = o a n (z)n3 n in Op because {an(zm)} uniquely determine {an(z)}. Thus we have an expression

20 (3a)

1: Algebraic number theory Op = \im(0/pm) and Z p = lim(Z/pmZ). m

m

It is easy to check that the topologies of both sides coincide (see [Bour2,1.4.4, III.3.3, III.7], [Bourl, IH.2.6]). In particular, this shows that (3b)

O^pmOp = 01 f1 and Zp/pmZp = Z/pmZ.

For any z e Op, we simply put z™ = 2^="oan(z)G3n e O. Thus we can always find an infinite sequence in O converging to a given z in Op; i.e. O is dense in Op. When O = Z , an(z) is always non-negative by our choice, i.e., z^ e N for all m. Thus (3c)

the set of natural numbers N is dense in Z p and O is dense in Op.

Exercise 2. Let q = #(O/p) for a (non-zero) prime p of O. Show that for an integer x in O, (i) {xqn}nGN is a Cauchy sequence in C^,, (ii) if x is prime to p, C,= limx qn is a (q-l)-th root of unity, (iii) if x is prime to p, | £_xqn4(q-i) | p < p-n a n ( i ^ a l l t j i e (q.i).^ r o o t s o f u n i t v ^ e obtained in this way. Naturally Op contains Z p for a unique prime p in Z such that pf]Z = pZ. Thus Op is the integral closure of Z p . The natural homomorphism of 0®zZ p to Op is surjective, because 0®zZ p is compact and its image contains the dense subring O of Op. Thus Op is finitely generated over Z p , and we can find a basis {wi, ..., Wf} (f 0, if xn = X ^i^x11"1 with aj e Op we have

ton+-+anG5m(n-1) whose right-hand side belongs to Op. Hence m has to be 0. This shows that x G Op and shows the normality of Op. In particular, Op is the integral closure of Z p . For this reason, Op is called the p-adic integer ring of ¥p. Let K/F be a finite extension. Let OK be the integer ring of K. For a non-zero prime ideal p of O, we decompose pOK = Tl^^ for prime ideals T of OKThus OK/fCk = n2>(OK/^e(5>)). Since as an 0-module, OK/2>= (O/p)m fora positive integer f(#), by (2.3b),

1.3. p-adic numbers (4a)

21

[K:F]

Then the ^P-adic completion R=OK,

and

K®FFP=YIT\P
Let E be a finite extension of ¥p and R be its p-adic integer ring (i.e. the integral closure of Op). Taking a basis {coi,...,cOd} of E/Fp, we easily see that G3aa>i € R for sufficiently large integer a. Thus FR = E. This implies that, if wi, ..., w r e R are O^-linearly independent, then r < d . This shows that R is free of rank [E:Fp] over Op. Since R = Op, R is p-adically complete. If p R =
R = lim(R//R) = fl

lim(R/2fR),

because R/p a R = n L ( R / ^ a e i R ) by the Chinese remainder theorem. Since R is an integral domain, s has to be equal to 1 and R is a valuation ring. By definition, if we write pR = & and f = dimo/p(R/(P), then (5)

[E:¥p] = ef and | a \T= \ a | f:F] for a e F.

Thus we can extend | | p to E by putting | x | p = \ x | ^ " -1 for x e E. The norm | | p is the unique norm extending | | p on F and defines the same topology as that given by | | #. We now introduce the p-adic exponential function exp and the p-adic logarithm function log for later use. To define these functions, we use the following power series expansions: _.n

(*)

/ 1 \n+l__n

exp(x) = X~=Q fj- and log(l+x) = £~ = 1 H ) _ J L .

By the strong triangle inequality, any power series f(x) = Z°l o a n x n is convergent at xe Op if and only if lim | anxn I p = 0. Here I I p is the normalized p-adic norm: | x | p = | x |

p

'

p

. Thus the radius of convergence is given by

R = (lim sup U n l J V (i.e. R is the largest real number which is a accumulation point of the sequence {U n l p 1 / n }). We know that | n ! |p /n = p-(1"(s/n))/(p-1) by (2) for the integer s = sn

22

1: Algebraic number theory

= Z°° a m (n)

for the finite p-adic expansion

n = L°° a m (n)p m .

m=0

Note that

m=0

I s | < (p-l)(l+log p I n I) for the complex logarithm logp with base p. Thus lim I — I = 0 and we have

n-»«> n

(6a)

The radius of convergence of exp with respect to I | p is p" 1 ^' 1 ^

As for the logarithm function, since I n | p < p"^logp'n' \ (6b)

The radius of convergence of log(l+x) with respect to | | p is 1.

Writing D(x,r) = {y e Fp \ | y-x | p < r}, we define exp : D(0,p"1/(p"1}) -> Opx and log : D(1,1) -» Op by the p-adically convergent power series (*). Let p be 4 if p = 2 and p = p if p > 2. Then for x e Z p with I x-1 | p < p \ log(x) e D(0,p"1/(p"1}), and we define xs = exp(s log(x)) for s e Z p . Using the formal identities in the power series ring, we can verify the following properties: (7a) (7b) (7c)

exp(x+y) = exp(x)exp(y) and log(xy) = log(x)+log(y), log(xs) = slog(x) for s e Zp, exp(log(x)) = x and log(exp(x)) = x,

Exercise 3. Give a proof of the above identities. If f(s) = 2T_ an(s-a)n is a power series converging around a e Op, then its formal derivative j-(s) = Z°°_1ann(s-a)n"1 also converges at a. This is clear from the inequality lim sup | an | ^ lim sup I (n+l)a n+ i | Writing f^n) for the formal n-th derivative of f, we have an = L^L

(8a)

, since

| n+11 p < 1.

for all n e N.

In particular, we see (8b)

^

=

Let Q be the p-adic completion of the algebraic closure Q p of Q p under | | p which is the unique extension of x

I l p on Q p . Let A = { x e Q

|x|p
x

For any x e Q , we see that x | x | p e A . Since Q p is dense in Q., for any positive n and any x e A x , we can find a finite extension K n /Q p and x n e K n such that

I x-x n |p < p" n . As seen in Exercise 2, co(xn) = lim x n p m ' is well

1.3. p-adic numbers

23

defined in Kn. Note that co(xn) is independent of n, for which we write co(x). Then we have where q = p[K«:Qp]. Thus

1 co(x)-xqn(q-1} | p < p"n. Then for x e A x , co(x) =

lim x1*1" is a unique root of unity with | x-co(x) | D < 1. We thus see that where

|i«, = {£ <= Q x | £N = 1 for some integer N prime to p}, p Q = { p r | r e Q} and r . o = { x e Ax | | x - l | p < l } .

The projection to Too written as X H ( X ) is given by © ( x l x l p ^ x l x l p ) . On Too, log is well defined. We then extend log to Qx by (9a) log(x) = log«x» (XG Qx). By definition, we still have (9b) logtxy"1) = log(x)-log(y). We conclude this section by giving a brief explanation of the Frobenius element. Let F be a p-adic field with p-adic integer ring O. We fix an algebraic closure F of F. Let O be the integral closure of O in F. We write p and p for the maximal ideals of O and O, respectively. We also write F = O/p and F = O/p, which is an algebraic closure of F. We note here that F is not padically complete [BGR, 3.4.3]. By definition, every automorphism of F leaves O and p stable and induces an automorphism of F over F. Thus we have a natural homomorphism of groups: p : Gal(F/F) -> Gal(F/F). The kernel of p is called the inertia subgroup, which we write I. Note that all non-zero elements of F are roots of unity. Then, it is easy to check that p is surjective by considering the extension F(Q for various roots of unity £ ([Bourl, VI.8.5]). There is a canonical generator n in Gal(F/F): rc(x) = x q for q = #(F). The element % is called the Frobenius element of Gal(F/F). If K/F is a (finite or infinite) Galois extension of F (see [N, Chapter I] for the Galois theory of infinite extensions) and if K is fixed by I, K is called unramified over F. If this is the case, on the residue field F' of OK, p induces a canonical isomorphism Gal(K/F) = Gal(F'/F). Thus in this case, we have a naturally specified element p4(7c) = Frob in Gal(K/F), which we call the Frobenius element of Gal(K/F). Now we suppose F to be a number field in a fixed algebraic closure Q of Q. We pick a maximal ideal p of the integer ring O of F. Let K/F be a (finite or infinite) Galois extension of F inside Q. We pick a maximal ideal T of PK over p, where OK is the integer ring of K. Then a e Gal(K/F) naturally acts

24

1: Algebraic number theory

on maximal ideals of OK- We denote by D = D((P/p) the stabilizer of <2 in Gal(K/F). This group D(

D(T/p) = Gal(Kp/Fp

([N, IV.l]).

We write l(T/p) for the inertia subgroup in D((P/p). If l(T/p) = {1}, we say fP is unramified over p. If all the maximal ideals over p in 0^ are unramified over p, we call p unramified in K/F. If Tip is unramified, we can consider its Frobenius element Frob(^P/p) in D(T/p). By (10a), we know that (10c)

Frob(2*7p) = aFrob(^P/p)a'1 if Tip is unramified.

Thus the conjugacy class of Frob(T/p) is well determined by p, which we write Frob r Sometimes, we identify Frob^ with F r o b ( ^ ) , which is actually a representative of the Frobenius conjugacy classes. We quote the following well known density theorem: Theorem 1 (Chebotarev). Suppose that only finitely many prime ideals in F are ramified in K/F. Then under the Krull topology on Gal(K/F) (see [N, LI]), the set of Frobenius elements Z = {Frob(ffl^) e Gal(K/F) | <Efp unramified in K/F} is dense in Gal(K/F). We do not prove this theorem here, because we need a large amount of either analytic number theory or class field theory to give a proof. We only refer for this to [N, V.6].

Chapter 2. Classical L-functions and Eisenstein series In this chapter, we will deal with basic analytic properties and some algebraic properties of abelian L-functions in a classical setting. To give an illustration of what we will do, let us start with a typical example of such results.

§2.1. Euler's method of computing ^-values In the following, a continuous C-valued function f defined in an open subset U of C is called holomorphic (or analytic) on U if f is differentiable on U and — = 0 for the variable z on U. By Cauchy's theorem, any holomorphic dz function on U can be expanded into a power series of z-u convergent absolutely on an open neighborhood of each u e U. A function defined on a dense open subset V of U is called meromorphic if there exist two holomorphic functions g and h * 0 such that f = f on V. The simplest example of an L-function is the Riemann ^-function given by the infinite sum for complex numbers s € C. C( s ) = X I - i n S This infinite sum converges absolutely if Re(s) > 1 and gives an analytic function in this region. In fact, it is clear that

I C(s) I < C(Re(s)) < l + J~x" Re(s) dx and the integral converges absolutely if Re(s) > 1. Moreover we shall find in §2.2 an analytic function f(s) defined on all s e C such that f(s) = (s-l)^(s) if Re(s) > 1. This function f(s) is necessarily unique by the Cauchy integral formula. The uniqueness can be shown as follows. If $ is an analytic function on the domain Q.T = ( z e C | | z | < r } , then by Cauchy's integral formula (see [Hor, 1.2]),

4>(s) = (2TI V z T)- 1 J | z | = t ^dz if | s | < t < r. Moreover we see that 36 3S

=

^ 4>(s+AsH)(s) As->0 As

26

2: L-functions and Eisenstein series

Here the interchange of the integral and the limit is possible because r

f(z-s-As)-1-(z-s)"1]

lim < As—»0 I

^g

converges uniformly on the circle ( Z G C I |z| =t}. In particular, we have —(0) = (2ft V—l) 3s Repeating the above process, we have

. ,
J|zl=t

Exercise 1. Give a detailed proof of the above formula. Since | s | < | z |, we see that — = z"x «- and | z-1s | < 1. Thus z s " l-z^s 1

^-= z" l-z^s —V ~ z s

+z"n-1sn+ Of course, this series is absolutely and uniformly convergent on any compact subset in Q r . Thus we have

= (2K V=T)"1J|z|=t = 0 identically. Thus if f(s) = (s-l)£(s) = g(s) on {s G C I Re(s) > 1} for two analytic functions f and g, we know that f = g on C. Now we write for the meromorphic function f(s)/(s-l). We will show here and in §2.2 that has a singularity at s = 1, Ress=i£(s) = 1; and also that £(s) has the following functional equation: (2TE)SC(1-S) " 2F(S)COS(TIS/2) '

Here F(s) is the Gamma function defined by J e'V^dt if Re(s) > 0 and satisfies F(n) = (n-1)! for positive integers n. Therefore, essentially the value of £(s) can be defined by the infinite series when either Re(s) > 1 or

2.1. Euler's method of computing £-values

27

Re(s) < 0. Now there is an interesting way to compute the value of £(l-n) for positive integers n, which was invented by Euler in 1749. We consider instead of the alternating sum: We want to compute (l-2m+1)£(-m) for m > 0. Euler's idea is to introduce an auxiliary variable t and consider

Then Euler pretended that the above series were convergent at s = -m and concluded by replacing t by 1:

(l-2m+1)^(-m) = f ^ T Y — 1 | t=1 for every integer m>0.

(1)

Here the right-hand side is the value of a rational function at t = 1 and hence a rational number. Thus if the above argument is correct, we have (2)

£(-n)

E

Q for n > 0.

Of course the above argument needs justification, but the result and the formula (1) are actually true. Exercise 2. (a) Show that the rational function

t—I f

J does not have

(t-1) as a factor in its denominator. (b) Explain why the argument of Euler is a little problematic. Of course, Euler was fully aware of the shakiness of his argument. Here is how he justified it. First he replaced t by e x . By the chain rule, we see that t-jrf(t) I

(3)

= T~f(ex) I _Q. Thus if you believe the formula (1), we have

(l-2m+1)C(-m) = QL)m( T f- r | | _ n for each integer m > 0.

Instead of x, we put 27cV~lz and write e(z) = e consider the function F(z) = ,

(/ = V - I ) . We then

/ y By (believing) (3), the Taylor expansion of

F at z = 0 is given by

(4)

F(z) = X (F(n)(0)zn/n!) = X ((l-2n+1)C(-n)(27CV^zf/n!). n=0

n=0

28

2: L-functions and Eisenstein series

By another formula of Euler, e*e = cos0 + V-IsinG, we know that iz , -iz

iz

-iz

cos(z) = —~—, sin(z) = From this fact,

cot(z) = V ^ l —

Therefore we have e(z)+l On the other hand, the function cot has the following partial fraction expansion: (5) For the moment, let us believe this expansion (which is absolutely convergent if z ^ Z). By the expansion of geometric series, we know, if | z | < 1 and z * 0, that

Then we see that

(6)

ei(z) = ^ + f ; £ n=lr=0

ln=l,k=l

J

k=l

Here only the terms for odd r survive and then we have written r+1 = 2k. Exercise 3. (a) Suppose that z g Z. Show that absolute convergence of E

{—— + —:} and also show that Z — z +— n is not absolutely convergent n=i z+n z-n (b) In (6), justify the interchange of the summations with respect to k and n (i.e. show rigorously the equality marked by ? in (6)).

From (6), we know that ! ; 2(l-22k)C(2k)z2k"1 (=} (VzT7i)-1(ei(z)-2ei(2z)) k=l

e(2z)+l e(z) 2 -2e(z)+l (e(z)-l)2 (e(z)-l)(e(z)+l) " ' e(2z)-l " " (e( _ 1 _ e(z) e(-z) e(z) , v( ? z

(

= -£ k=l

2(l-22k)C(l-2k)(27cV::Tz)2k-1/(2k-l)!.

2.1. Euler's method of computing £-values This shows that

29

C(2k) =

By specializing the functional equation £(s) =

at s = 2k, this 2r(s)cos(ics/2) equality is in fact true, because cos(kTt) = (-l) k and F(2k) = (2k-l)!. At the time of Euler, the functional equation was not known and in this way, Euler predicted its form. To make sure of our logic, we summarize our argument. Introducing an auxiliary (d y^f ex ^ I variable t, Euler related the value I —I - — r | _0 with £(-m); so we write (l-2 m+1 )a m for this value, which is not yet proven to be equal to (l-2 m+1 )£(-m). Then by definition, the formula (4) read F(z)=n=0

On the other hand, by using the partial fraction expansion of the cotangent function, we computed the power series expansion in (6): ^

e(z)+l_l__

+i Since F(-z)-F(z) =

(

T+"iz

k=i

fl , - 2 n \ i > equating the power series coefficients of

the two sides using the above two formulas, we obtain Thus we know Proposition 1.

£(2k) e 7C2kQ for all 0 < k e Z.

On the other hand, by specializing the functional equation (which we have not proved yet) at s = 2k, we have C(2k)= Then we conclude, assuming the functional equation, that

C(l-2k) = a*.! = (l-2 2 Vfxf

"TTT>

I I x=0-

Here are some remarks, (a) By the formula (1), one can compute C(2k) or £(l-2k). Here are some examples: C(2) = 1 + 2 - 2 + 3 -

2

+-

= \ , ....

30

2: L-functions and Eisenstein series

For more examples, see the table given in [Wa, p.352]. (b) The primes appearing in the numerator of £(2k), for example 691, are called irregular primes and have arithmetic significance. In fact, the class number of the cyclotomic extension Q(CP) for a root of unity £p with £pp = 1 but £p * 1 is divisible by p if and only if the prime p appears in the numerator of C(l-2k) for some k with 2 k < p - l . This is the famous theorem of Kummer proved in the mid 19th century and immediately implies the impossibility of a non-zero integer solution to the Fermat equation xp + yp = zp if p is regular (i.e. not irregular). We refer to [Wa] for more details of this direction of research and to [Ri] for the approach using modular forms. (c) The nature of the value £(2k+l) for a positive odd integer is quite different from the even values £(2k) and they are supposed not to equal a rational number times a power of n. We insert here a sketch of the proof of (5) due to Eisenstein and Weil [W2, II]. We shall show that the right-hand side of (5) satisfies the differential equation y' = -y2-7i2. The solution of this equation which goes to °° at z = 0, as is easily shown by a standard argument, is unique and equals Ttcot(Ttz). We put l v-»~ r 1 1 ^ >* + + lf r = 1 if r > 1. de These series are absolutely convergent. Here note that - p = -re r+ i. Taking two

z

X

fe

z^

independent variables p and q and putting r = p+q, we get — = — + —. pq pr qr

a

a

Differentiating once by — and —, we get (keeping the fact that r = p+q in dp dq mind) _1 1_ J_ _2_ _2_ _1_ J _ _1 2_ _2_

pV

=

PV

+

qV

+

pr3

+

qr3

Or

p V " PV " q V

=

pr3

+

qr3 '

In this equality, we put p = z+n, q = w+m-n with integers m and n. Then r = z+w+m and summing up with respect to n keeping m constant, we have

r- 3 X n {jU 1-} = 2(z+w+m)-3{e1(z)+e1(w)}. Now we sum with respect to m: (7) e2(z)e2(w) - 82(z)e2(z+w) - e2(w)e2(z+w) = = 2{ei(z)+ei(w)}]Tm(z+w+m)-3 = 2e3(z+w){ei(z)+£i(w)}. Differentiating (6) with respect to z (noting the fact that —- = -e 2 ), we get QZ

e 2 (w) = w"2 + Y~=i (2r-l)2^(2r)w2r-2 = w"2+ 2£(2) + 6i;(4)w2 + —

2.1. Euler's method of computing £-values

31

Now expanding £2(z+w) into a power series in w at w = 0 regarding z as a constant, we have, by the formula —L = -re r + i, e 2 (z+w) = X ~ o (e 2 (r) (z)w r /r!) = £ ~

0

(-l)r(r+l)er+2(z)wr,

£ 3 (z+w) = X r °l 0 (-Dr(r+l)(r+2)er+.3(z)wV2. Then the constant term of the left-hand side of (7) in the power series expansion with respect to w is given by 2C(2)e 2 (z) - £ 2 (z) 2 - 3£4(z) - 2£(2)£ 2 (z) = - £ 2 ( z ) 2 - 3e 4 (z). The constant term of the right-hand side is (by using (6)) 2£3(z)£1(z)-6£4(z). Thus we have (8a)

3e4(z) = e 2 (z) 2 + 2e 3 (z)ei(z).

Similarly by expanding both sides of (7) into a power series in x = z+w at x = 0 regarding z as a constant and equating the constant term, we have (8b)

£ 2 (z) 2 = £4(z) + 4C(2)£2(z).

Eliminating £4 using (8a,b), we know that (8c)

£i£ 3 = £ 2 2 - 6£(2)£ 2 .

Differentiating this formula, we know from —L = -r£r+i that QZ

(8d)

£ 2 £ 3 - 4^(2)£ 3 = £i£ 4 .

Multiplying (8b) by £i and eliminating £i£4 using (8d), we have ei£22-4^(2)£i£2 = £2£3-4C(2)£3 or equivalently £2(£3-£i£2) = 4£(2)(£3-£i£2). Since £2 is not the constant 4^(2), we know that £3 = £i£2. In (8c), we replace £3 by £i£2 and then divide by £2 to obtain 2 = £ 2 - 6C(2). £l Then by the facts - p = -62 and 6£(2) = n2, we know that £1 satisfies the differential equation y1 = -y2-7C2. The fact 6£(2) = 7c2 will be shown independently of this argument. Now we give a generalization of the formula (1):

(^J^) t=i

(l-2m+1)C(-m) = ( t ^ J * ^ ) I t=i for each integer m>0, which was found by Katz [Kl]. Instead of 2, we fix an integer a > 2. We define a function \ : Z —> Z by 1 if n gfe 0 mod a, 11-a if n = 0 mod a.

32

2: L-functions and Eisenstein series

We note that

Y^l^{b)

= £j = 1 5(b) = 0. We consider, instead of j ^ , the

rational function O(t) = — 7 ~ a i — • Then we see that t—l t —1
t+l

2 By computation, we see that O(t) =

t22 + l

(t+l) 2 -2(t 2 +l)

?T

£i

(t+Dd+t+tV-'+t^-ata-a l-a+2(t+t2+---+ta"1)ta-l ta-l 2l _ 1 ^(b)(l+t+t2 +-+tb - 11) a

Now we put

S L ^

T(t) =

^ ( b ) ( t t t ) I ^(b)t , pj - = b - l S aa . a 1 l+t+r+-+t " l-t

1

The last

equality follows from Z^ =1 ^(b) = 0. When a = 2, we have ^ ( t ) = J^T. In -2vP(t) = 3>(t)-(a-l). Then by the formula which we have

general, we have already looked at,

*(e(z)) = -(i7i)- 1 X~ =1 2(l-a 2k )C(2k)z 2k -\ we now know that

To relate this formula with the values of C, at negative integers, we consider

^(n)tn, we get

Putting g(t) = Z n=l

( d^ m and hence formally

(•5 On the other hand, we have

2.2. Analytic continuation and the functional equation d^ m

= [t^j

33

tna > . I t=l

g(t )

z=0

This formal computation can be justified by applying a functional equation (to be proved in the following section) to the right-hand side of (l-a 2k )Cd-2k) = • Thus we obtain Theorem 1. Let a > l

be an integer.

Then for each positive integer

m,we

b

( A \m . Zu_iS(b)t (l-a m+1 )C(-m) = t^- Y(t) I t , w/z^r^ ^F(t) = b - 1 . .

have

\ oij

t=

i

1-t

Exercise 4. We have only proved the theorem when m is an odd positive integer. Give a proof of the theorem for even positive integers m. (First show d^ Wf e — -—5f n = 0 for 0 < m £ 2Z by making the substitution X H - X dxy ^l + e j • x=o and then show £(-m) = 0 by using the functional equation.) Exercise 5 (the Lipschitz-Sylvester theorem). By using Theorem 1, show that a m+1 (l-a m+1 )£(-m) e Z for every integer a> 1, where m is a non-negative integer. Exercise 6. (a) For each prime p, show that (Z/pZ)x is a cyclic group, (b) For a positive even integer k, show that if the denominator of £(l-k) is divisible by a prime p, then k is divisible by p-1.

§2.2. Analytic continuation and the functional equation As seen through Euler's argument, we now know the importance of the function F(z)

=

i , / \ in the theory of the Riemann zeta function £(s). Here we modify

this function a little to deal with £(s) directly (instead of (l-2"s)£(s)). Put e"z G(z) = —— for z G C. First we give an integral expression for £(s). For i~e that, we need several properties of the F-function. Let us recall some of them. We may take the following integral as a definition of the F-function:

(1)

F(s) = Jo~ e-y-My if Re(s) > 0.

34

2: L-functions and Eisenstein series

This integral is convergent only when Re(s) > 0. To show this, we remark that for a fixed real number e > 0 and for any real number a, there exists M > 0 such that if y > e, then My"2 > e^y0"1. Thus writing a = Re(s), we see that

I r(s) I = I Jo~ e-V-xdy I * J~ I e-V 1 1 dy = J~ e V ' d y . In particular, we know that

I JEf e-VMy I <: Je. f e V ' d y < MJE[~ y-2dy = M[-y"T = Me"1. £ Thus the integral J e"yys"1dy converges for all s and gives an analytic function of s. The problem of divergence lies in the other integral:

I Jo£ e^y-My I < J* e'V^dy < J* y^dy < [ya/a]* = ea/a if a > 0. To find the analytic continuation of F(s), we consider eyys~l as a function of the complex variable y. Here note that the function y — i > ys is not well defined s slog because y = Q W and log is multivalued. To fix a branch, we write /G with 0 < 9 < 2TC and define log(y) = log \ y \ +/0 and y = | y | e s/ s y = e ^(y). When 0 = 0 or 2K (i.e. y is on the positive real axis), we write ys when 0 = 0 and y.s when 0 = 2n. Note that y11 = y.n for integers n. We fix a positive real number e and denote by 3D(e) the integral path which is the circle of radius e with center 0 starting from e = I e I e l0 and with counterclockwise orientation, by P+(e) the path on the real line from +°o to e and by P_(£) the path on the real line from e to +<*>. We consider that 0 = 0 on P+(e) and 0 = 2% on P_(e). We write the total path as P(e) = P+(e)U3D(e)UP-(e). We similarly write 9D(e,ef) for the boundary of the annulus cut along the real axis: {P+(ef)-P+(e)}UaD(e)UaD.(eI)U{P-(e')-P-(e)} for e > e' > 0, where 3D.(e') = 3D(e') as a path but has the clockwise orientation: Then

j P + ( e )e" y y s " 1 dy

and

Jp (£)e~yy-s~1(ty converge for all s and give analytic functions on C. Note that

2.2. Analytic continuation and the functional equation e y S ld +(e) " y " y

35

= - Je~ e-V'dy and / p ^ e ' V ^ y = e2ltis J£" e ^ d y .

Changing variables by y = ee l9 (dy = *ee'ed9), we have, for a sufficiently small e,

I JaD(E)e"yyS"ldy I ^ Me a J^dG = 27iMe a if a > 0, where M is a constant independent of e. Thus if a = Re(s) > 0, we know that

^ o W ' V ^ d y = (e27t's-l)r(s). The function e^y**"1 has no singularity on 3D(e,e') and hence, by Cauchy's integral formula (see [Hor, 1.2]),

Zz:poles ta v^^-yf-1

f y y

= o.

Then we have

Thus we know that the holomorphic function Jp(8)e"yys"1dy of s is independent of £ and gives (e 27ns -l)r(s) if a > 0. Thus we have the meromorphic continuation of F(s) given by

(2)

r(s) = (e2ms-l)-1Jp(E)e-yys"1dy for all s e C.

Since (e271^-!)"1 has singularities only at integers s e Z, which are simple poles, F(s) has at most simple poles at non-positive integers. Through integration by parts, we have the well known functional equation

r(s+l) = Jo~ e'Vdy = [-e"yys]Q+s Jo~ e ' V ^ y = sr(s). The analytic continuation of F(s) can be also proven by using this functional equation, which shows that F(s) has in fact simple poles at integers m < 0. Exercise 1. Show that for each positive integer n

Now we go into the integral expression of £(s) using G(z). Expanding G into a geometric series, we know that

(3)

G ( y ) = - ^ = £ e"ny. X

This series is convergent when integrate G on R+:

"e

n=l

| e' y | < 1 (<=> Re(y) > 0). We first formally

36

2: L-functions and Eisenstein series

n=l

n=l

n=l oo

X n=l

/.OO

and I

made at the equality marked "V

by "?". For that, we look at the poles of G(y) =——. Since l-e' y has zeros x-e only at 2%4-irL = {27cV-Tn|nG Z}, G(y) can have a pole only at 0 on R. Exercise 2. Show that | yG(y) I is bounded (independently of y) in the unit disk of radius one with center 0 (hint: show lim yG(y) = 1 and deduce the result from this). On the other hand, since e"y -» 0 as y -> +©°, there is a constant M > 0 such that I yG(y) I < Me" y/2 by the above exercise. This shows first of all, for a = Re(s),

I J~ G(y)y s " 1 dy I < Jo°° I G(y)y I y°-2dy < M Jo°° e" y/2 y a ' 2 dy = 2°-iMr(a-l). Thus

I G(y)y

dy is convergent if a > 1. Since the domain of integration

R+ is not compact, the uniform convergence of (1) is not sufficient to get the in(•OO

terchange of Z°° n—1

and

. In order to assure the interchange, we shall use the JQ

following dominated convergence theorem in the integration theory (due to Lebesgue): If a sequence of continuous (actually integrable) functions fn(x) (on an interval [a,b] in R; a and b can be ±<») is dominated by a continuous and integrable function and f(x) = lim fn(x) at every point x, then rb

rb

lim fn(x)dx = lim

fn(x)dx.

We apply this result to Gn(y)ys"1 = E ^ e ' ^ y 8 " 1 which is dominated by the integrable function

I G(y) | y0"1 (if a > l ) and converges to E

e'^y 8 " 1 . In

n=l

fact, we see that

I GnCyV11 = | I ^ ^ ^ V 1 1 * S; =1 h" my y s "i = I G(y) I y0"1 if y > 0. Therefore we have

j ; - " y y - M y = lim j;Gn(y)dy= j ; UmGn(y)dy= f f n=l

n=l

2.2. Analytic continuation and the functional equation

37

This justifies the interchange and we have Proposition 1.

F(s)C(s) = J~ G(y)ys"1dy if Re(s) > 1.

We cannot extend (naively) this integral expression to the left half plane { s | R e ( s ) < l } because £(s) has a simple pole at s = l. Since G(y) has singularities only at y = 0 in a small neighborhood of R+, we know, in the same manner as in the case of the analytic continuation of F(s), that if 0 < e < In and Re(s) > 1,

r(s)C(s) = Jo~ G(y)ys"1dy = (e 27l/s -l)- 1 J P(e) G(y)y s - 1 dy. The integral of the right-hand side converges for all s and gives an analytic function of s. Thus we have as a corollary of the proposition. Corollary 1. (e27ns-l)F(s)£(s) can be continued to a holomorphic function on C and has an integral expression: (e 2 ™-l)F(s)C(s) = J P(e) G(y)y s - 1 dy for 0 < e < 2TC. Exercise 3. (a) Prove Corollary 1 rigorously along the lines of the proof of the analytic continuation of F(s). (b) Compute Ress=iC(s) by using the above integral expression. We now want to compute the value of Jp(e)G(y)ys"1dy at s = 1-n for positive integers n. As in Exercise 1, we know that

We define the Bernoulli numbers B n by

We see that B n = (— J{xG(x)} I x=Q. Since Resy=0G(y)y"n = ^ f , we know that "n^v

=

(27cV—T)—p. Then by Exercises 1 and 4(b) below, we know p

Theorem 1. For each positive integer n, we have £(l-n) = — - if n is even, n G(0) = - i and C(l-n) = 0 if n > 1 is odd. This agrees with Euler's computation via Exercise 1.4.

2: L-functions and Eisenstein series

38

Numerical Example. We list here several Bernoulli numbers: Bo

=

o7

23'

" 4 2 "2.3-7' * 1 0 ~ 66 "2.3-11'

30

691 691 "2730 ""2.3-5.7.13' (See [Wa, p.374] for more Bernoulli numbers). Exercise 4. (a) Show that B2n is positive if n is odd and B2n is negative if n is even (Use the functional equation of £(s)). (b) Show that B n = 0 if n is odd and n > 1. (c) Show that the denominator of B n consists of primes p such that n is divisible by p-1 (use Exercise 6(b) in the previous section). Now we prove the functional equation. The idea is to relate the function of s given by Jp(e)G(y)ys"1dy with the integration of G(y)y's"x on the following integral path for each integer m > 0: (2m+l)
We denote this integral path by A(m), where the orientation is clockwise on the outside rectangle. Since the orientation of A(m) is clockwise if you look from a point inside of A(m), by Cauchy's integral formula, we know that

-2m-l

A(m)

where Resy=w(|)(y) is the coefficient of (y-w)"1 in the Laurent expansion -oo c ( n )(y- w ) n function G(y)ys"1 =

_y

Since

l*~y = 0 if and only if y e 2% V^IZ, the

^ " 1 has simple poles only at 27cV-ln for integers n x~e withO< |n I < m inside A(m). The residue at y = 27cV-Tn can be easily computed and is given by /^/xsu I -J-i\2Mi\s-\s%il2 ifn>0, (4) Res^GWy-)!

2.2. Analytic continuation and the functional equation

39

Exercise 5. Write down every detail of how one obtains the formula (4). Thus we have n=l

Therefore, if Re(s) < 0, we already know that

lim L^n 55 " 1 converges to

£(l-s) and hence we obtain (5)

i nmJ A ( m ) G(y)y

s 1

- dy = (27c)sems(e7lis/2-e-7Cis/2)C(l-s) if Re(s) < 0.

On the other hand, if we denote by Q(m) the square of side length 4m+2 centered at 0 with clockwise orientation and by P(e,m) the complement of Q(m) in A(m), then we have m HmJ A(m) G(y)y

s 1

- dy = ta y)y s - 1 dy +

JimJ Q(m) G(y)y s - 1 dy We now show that lim Jo(m)G(y)ys"1dy = 0. Then after a minor modification, we will have the functional equation. We write Q±(m) for the upper and lower edge of Q(m) and Qr(m) (resp. Q/(m)) for the right (resp. left) edge of Q(m). Here we only prove that

The computation for the other edges will be left to the reader as an exercise. First note that, writing y = X+TCV-I (2m+l) on Q+(m) for x = Re(y), we have e-y = e- 7U e" 2m7 V x = -e"x, I l-e" y I = l+e" x > e"x and |G((2m+l)7cVz:l+x)((2m+l)7cV=:r+x)s-11 < M | (2m+l)7cV :: l+x| a - 1 for a constant M depending only on s and for a = Re(s) < 0. Hence, we have

I G(y) | = -^—- < 1 for all y e Q+(m). This shows that 1+e

|G((2m+l)7tV :: r+x)((2m+l)jcV :: T+x) s - 1 |dx —2m—1

f —2m-l

40

2: L-functions and Eisenstein series

If a < 1, then o-l < 0 and we see that ((2m+l)V+x 2 ) ( a - 1 ) / 2 Then we have

f2m+1

•—2m—1

*—2m—1

which goes to 0 as m ^ +°o if a < 0. This shows the assertion. Exercise 6. Show that lim Jgr(m)G(y)ys"1dy = 0. (First show that on Qr(m), I G(y) I < 2 and proceed in the same manner as above.) By the above estimate and Exercise 6, we obtain (e27n's-l)r(s)C(s) = (27i)sem's(e7C/s/2-e-^/2)!;(l-s) or multiplying by e~ms, we have = (27i)s(e7Cfs/2-e-m"s/2)C(l-s). e 7Us/2_ e -ms/2

By the fact that J

:

:— =

JKIS _ -7WS

%$

{2COS(-TT)} l

Theorem 2 (Riemann). We have

§2.3.

V

2

, we deduce

''

£(s) = (27C) ^ 1 " s ) 2r(s)cos(7is/2)

Hurwitz and Dirichlet L-functions

Now we extend our argument to Dirichlet L-functions. Let % be a character of the multiplicative group (Z/NZ)X. Thus % : (Z/NZ)X -> C x is a homomorphism of the finite multiplicative group (Z/NZ)X into C x . We extend % to a function on Z by putting %(n) = %(n mod N) if n mod N is in (Z/NZ) X and %(n) = 0 if n mod N is outside (Z/NZ)X and define the Dirichlet L-function of X by Any x e (Z/NZ) X satisfies xr = 1 and hence %(x)r = 1, where r = cp(N) is the order of (Z/NZ)X. Thus %(n) is a root of unity and in particular satisfies I %(n) I < 1. It is an algebraic integer of a cyclotomic field. Then we have

\L(s,x)\ * XI=i 15C(n)ns | < J^=1 n"° = ^ ) for a = Re(s). This shows that L(s,%) is absolutely convergent if Re(s) > 1 (this convergence is uniform on every compact subset in the region with Re(s) > 1 and hence L(s,%) is an analytic function of s in this region).

2.3. Hurwitz and Dirichlet L-functions

41

Exercise 1. Show the following Euler product expansion of L(s,%) and its convergence if Re(s) > 1: L(s,%) =

U

p

(

l

More generally we think of a function c|>: Z/NZ —> C and consider

To continue this function to the whole complex plane, we rewrite it as oo

N

N

L(s,§) = ^<|)(n)n~s = ^(()(a)^(Nn + a)~s = ^(|)(a)N~ s ^(n-l-—)~ s . n=l

a=l

n=0

a=l

n=0

Thus the problem of analytic continuation of L(s,<|)) is reduced to the same prob(n-hrj)'s. More generally we consider the following zeta function for

lem for X n=0

-N

all 0 < x < 1:

£(s,x) = £ ~ = 0 (n+x)-s, which is convergent if Re(s) > 1. This zeta function is called the Hurwitz L-function (and was introduced by Hurwitz in the 188O's). We then have

In the same manner as in the case of the Riemann zeta function, we have the following integral expression of £(s,x): (2)

r(sK(s,x) = J~ G(y,x)ys"1dy if Re(s) > 1,

where G(y,x) = ^ y - ^ - = - ^ y

= £~=Q

e-

(n+x)y

.

Exercise 2. (a) Show that £(s,x) is absolutely convergent if Re(s)>l. (b) Show the formula (2) along the lines of the proof of Proposition 2.1. Note that the zeros of l-e' y are situated at 27iin for n € Z and are all simple. In particular, yG(y,x) is bounded in any small neighborhood of 0. Thus for sufficiently small e > 0 (actually, any e with 0 < e < 2n does the job), the integral Jp/8xG(y,x)ys"1dy is convergent for all s and gives an analytic function of s. The same computation as in the proof of Corollary 2.1 gives Proposition 1. (e27liS-l)F(s)^(s,x) can be continued to a holomorphic function on C and has an integral expression: (e 27U ' s -l)r(s)C(s,x) = J P(e) G(y,x)y s - 1 dy for 0 < e < 2TC.

42

2: L-functions and Eisenstein series

This combined with the formula (1) yields Corollary 1. The function (e27US-l)r(s)L(s,(|)) of s can be continued to a holomorphic function on C and has an integral expression: -i d y for

0 < e < 27C/N.

As a byproduct of the analytic continuation, we can compute the value of £(l-n,x) using the formula {(e27C/s-l)r(s) | s=1.n}Cd-n,x) = JaD(£)G(y,x)y-ndy = (2TC VzT)Resy=oG(y,x)y-n. By Exercise 2.1, we know that

(e 2 ™ D H s ) I i;i

(e

Wi

( 1)n 1(27cV=II)

'

"

n

Thus we compute Resy=oG(y,x)y" . Write F(y,x) = yG(y,l-x) and expand F(y,x) into a power series in y (regarding x as a constant):

We can interpret this argument in a manner similar to §1 as follows. Writing tx y 11

t = e , we have F(y,x) = yfy. Thus pr = - + Z°° / t n t y - This shows i~ x

i~ x

^ - a-f^ = £^(B n + 1 (x)-a^B n + i(x)) y n

(1 a

"

}

n+i

n^O \n~T" x) *

y

for a n y i n t e g e r a > 1? a n d t h u s

- I'dTj \t-rV-ij I '=!•

Then Bn(x) is a polynomial in x of degree n with rational coefficients. These are called "Bernoulli polynomials". We can make this more explicit as follows:

Therefore, equating the coefficients in y11, we see that Bn(x)

=

yn

n! ^ In other words, we have the formula

j=o

B

j

x n-j

j!(n-j)!

Bn(x) = ]T*=o (^Bjx 1 1 ^ G Q[x].

(3b) Thus we know that

Res y=0 G(y,x)y- n = Res y=0 F(y,l-x)y- n - 1 = the coefficient of y11 in F(y,l-x) = This shows that

Bn(1

"x). n!

2.3. Hurwitz and Dirichlet L-functions

(

)

;

j

43

j

Thus we conclude that £(l-n,x) = (-1)11"1 n^ (l-x)y

. Here we note that

e -xy

This equality yields B n (l-x) = (-l) n B n (x).

(4) Thus we obtain

Theorem 1. We have, for any integers a > 1 and 0 < b < N,

N»(l-,~«K(-m|) - ( . i y ^ - . J * , ) | „, for allra20, and

Then the formula (1), L(s,<|>) = E^=1(|)(a)N"sC(s,^), combined with Theorem 1, implies Corollary 2. Let Q(%) denote the cyclotomic field generated by the values of the character %. Then we have Moreover, suppose %(-l) ^ (-l) n . ^/zen we have L(l-n,%) = 0 /or n > 0 if % is non-trivial, and £Q- n ) = 0 if n > 1. In general, for ty : Z/NZ --» C, we have, if SL is prime to N, L(-m,Hm+1W = fffffyWt" a f ^ f ] I t , /or a// m > 0, dtJ t=1

^

l&T^ ST^^J '

The vanishing of L(l-n,%) follows from (4) if %(-l) * (-l) n . The number Bn>% = Z asiXteJN^Bnfe) is called the generalized Bernoulli number. Using the notation

Bn>5c, the above formula takes the following

Examples of Bernoulli polynomials:

B0(x) = 1, Bi(x) = x-i , B2(x) = x 2 - x + \ , B3(x) = Using the above formula, we can get X

^ ^

if

X is non-trivial.

form:

44

2: L-functions and Eisenstein series

Since % : (Z/NZ) X -» C x is a group character and (-1)2 = 1, %(-l) 2 = 1. Thus % ( - l ) = ± L Now we prove the functional equation of L(s,%). We proceed in the same way as in the case of the Riemann zeta function. We consider the integral on the path A(m) for each positive integer m given in §2. By the Cauchy integral formula,

The function G(y,x) has a simple pole at 2ft V—In for integers n and the residues at 27iV—In can be computed as ,„,

NsK

Resy=27t(n(G(y)x)y

U

«

because as already computed, Resy=2mn c-2ninx+(s-l)ni/2

| 2n% | s-1

>

W

I2nn | ^

) = je.27tinx+3(s.1)7li/2, ^

Qr

z=l l-e~y

e-27imx+3(s-l)7rf/2

if n > 0 ,

,s ,

.{n
and the value of &'xyysA is

| 2n% | s-1

a c c o r d i n g aS

n > 0 Or

n < 0. Therefore

This shows, if Re(s) < 0,

JmJ A(m) G(y,x)y s - 1 dy = In exactly the same manner as in the case of the Riemann zeta function, the integral on the outer square Q(m) converges to 0 as m —> +°°. Thus we have

(e 2 *' s -l)r(s)C(s ; x) = J P(e) G(y,x)y s - 1 dy = |KmJA(m)G(y,x)ys-1dy = (27t) s {e 3nis/2 Y~

e 2xmx n s-l_ e ms/2y~

-2xmxns-lj

i

s

Then by the formula L(s,<|>) = I^=1<))(a)N" ^(s,^), we see that (e 27t ' s -l)r(s)L( S) %)

where e(z) = e27liz. Now we need to compute Xa=i %(a)e(na/N). To treat this sum, we insert here a definition. For each proper factor D of N, we have a natural homomorphism p D : (Z/NZ) X - > (Z/DZ) X which satisfies p D (n mod N) = n mod D. If for some D, there exists a character %o • ( Z / D Z ) X -> C x such that %(n) = %O(PD(H)) for all n e (Z/NZ)X, we say % is imprimitive. A character % is called primitive or primitive modulo N if % is not imprimitive. Now we suppose

2.3. Hurwitz and Dirichlet L-functions (5)

45

% is a primitive character modulo N.

We do not lose much generality by this assumption: Exercise 3. Let % = %O°PD for a proper divisor D of N. Then show Ms,X) = {n p | N (l-%o(p)p" s )}^(s,Xo), where p runs over all prime factors of N. First we prove the following lemma (the orthogonality relation) in group theory: Lemma 1. Let G be a finite abelian group and % : G -> C x be a character. If % is not identically equal to 1, then Xg^GX(g) = 0. Similarly if g ^ 1, X%%(S) = 0» where % runs over all characters of G. Proof. Since % has values in the group of roots of unity, which is cyclic, we may assume that G is cyclic and % is injective by replacing G by G/Ker(%). Let N be the order of G. Then % induces an isomorphism of G onto the group |IN of N-th roots of unity. Thus X*(g) = X s = 0 because n^e^ N (X-C) = X N - 1 . geG

;ey. N

Let G* be the set of all characters of G. Defining the multiplication on G* by %V(g) = %(g)V(g)» G* is a group. If G is cyclic of order N, then the character is determined by its value at a generator go. Thus G* s % h-> %(go) e (IN = {£ e C x I £ = 1} defines an injection. For any given £ e [i^, defining %(gom) = £m> % is a character having the value £ at go- Thus G* = (IN- Then assigning g the character of G* which sends % to %(g), we have a homomorphism: G -> G**. Since %(g) takes all the N-th roots of unity as its values at some %, this map is surjective. Then by counting the order of both sides, we conclude that G = G**. In general, decomposing G into the product of cyclic groups, G* will be decomposed into the product of that character groups of each cyclic component. Thus G = G** in general. Then replacing G by G* and applying the first assertion of the lemma, we get the second. Lemma 2. Define the Gauss sum G(%) = Z^=1%(a)e(a/N). Suppose that % is primitive modulo N. Then ^

%(a)e(na/N) = %~l(n)G(%) for all integers n.

Note here that %~l(n) = %(n)"1 is again a character of (Z/NZ)X, which is primitive. In particular, this implies Za=1%(a)e(na/N) = 0 if n g (Z/NZ) X because %"1(n)=0 by our way of extending %A outside (Z/NZ)X.

46

2: L-functions and Eisenstein series

Proof. Define \|/ : Z/NZ -> C x by \j/(t) = e(t/N). Then, \|/(t) does not depend on the choice of t. In fact, if t = s mod N, then t = s+Nn for an integer n and thus e(t/N) = e((s/N)+n) = e(s/N)e27l/n = e(s/N). Thus \|/ gives a homomorphism of the additive group Z/NZ into the multiplicative group C x . This fact is obvious because \j/(t+s) = e(t+s) = e 27lf(t+s) = e 27lft e 27C/s = \|f(t)\|f(s). Then S^=1%(a)e(na/N) = S ae(Z/NZ)X X(a)\|/(na). We first treat the case where n mod N e (Z/NZ)X. Then the multiplication of n induces a bijection x h-> nx on (Z/NZ)X. Thus we can make the variable change in the above summation; so, rewriting na as a, we have

£ %(a)\|/(na) = ae(Z/NZ)

x

^ ( n ^ a ^ a ) = %l(n) a€(Z/NZ)

x

£x(a)\|f(a) = X'l(n)G(x)-

ae(Z/NZ)x

Now assume that n £ (Z/NZ)X. In this case %"!(n) = 0 by our way of extending x"1 outside (Z/NZ)X. Thus we need to prove that

Let p be a prime which is a common divisor of N and n. Write N = pD and n = pnf. Then we have

£x(a)\j/(na) = ae(Z/NZ)x

]T%(a)e(n'a/D) = ae(Z/NZ)x

I^n' ae(Z/DZ)x

because e(n'a/D) only depends on the class of n'a modulo D (but not N). We shall show that S t e a m o d D%(b) = 0. We see that

X

E

1

X

D%(b)«

Xb=a mod D%(b) = Eb^a mod D % ( a b a ) = X( a )Xb.l mod

Let H = ( x 6 (Z/NZ) X | x = 1 mod D). Since H = Ker(p D ), it is a subgroup of (Z/NZ)X. If % is trivial on H, then we define %0 : (Z/DZ) X -> C x by %O(PD(C)) = %(c). Xo is well defined because if PD(C) = PD(C'), then c = c'h for h G H = Ker(p D ). Thus %(c) = %(c'h) = %(c')%(h) = %(c') because of the triviality of % on H. Then % = %O°PD> which contradicts the primitivity of %. Therefore we can conclude % is non-trivial on H. Thus the orthogonality relation of characters (Lemma 1) shows that

2.4. Shintani L-functions

47

Now by using this lemma, we finish the computation:

(e27tis-l)r(s)L(s,x)

We see easily that c2nis-l e37t;s/2_x-i(_1)e™s/2

if x ( - l ) = 1, if %(-l) = - 1 .

|2COS(TCS/2)

{2^pLsin(ns/2)

This shows Theorem 2. Suppose that % is primitive modulo N. Then we have 'G(%)(2TC/N) S L(1-S,%- 1 )

^>3"

2r(s)cos(7is/2) ^ G(x)(27r/N)sL(l-s,%-1) . 2Vzir(s)sin(7Cs/2)

.,

if

^

Exercise 4. Using the above functional equation, show that L(s,%) is a holomorphic function on the whole complex plane C if % is a primitive character modulo N > 1. (The main point is to show that L(s,%) is holomorphic at s = 1; use also Corollary 2.) Exercise 5. Suppose that % is primitive modulo N. By using the functional equation (and also the power series expansion of L(s,%) at s = -j), show G(x)G(%"1) = %(-l)N for general primitive %, and supposing that L(w,%) * 0, show

G(%) = ^3C(-1)N

if % has values in {±1}. (The fact that G(x) = has values in {±1} is true without the

assumption of the non-vanishing shing of L(s,x) at ^- Try to prove it without the non-vanishing assumption.) §2.4. Shintani L-functions In this section, we introduce the contour integral of several variables and Shintani L-functions [Stl-6] and later, we will relate them with Dedekind and Hecke L-functions of number fields. We now take another branch of log different from the one in the previous section; namely, for z e C, we write it as z = | z | tlQ

48

2: L-functions and Eisenstein series

with -K < 0 < n using the polar coordinate and define log(z) = log I z | +/0. Accordingly, we define the complex power zs = es/o^(z^ by this logarithm function. We put H ' = { z e C | Re(z) > 0}: the right half complex plane, R ± = j x e R | i x > 0 ) : the right or left real line, R ± = R ± U{0}, N = By our choice of log, we have the luxury of (ab) s = a V , as = a ? and (a 1 ) 8 = a"s for any two a,b

G

H' and s e C.

To define the Shintani L-function, we need the following data: (i) a complex rxm matrix A = (ay); (ii) % = (%i, ..., %r) e C r with |%il £ 1 for all i; and (iii) x = (xi, ..., xr) G R r such that 0 < Xi < 1 for all i but not all Xi are 0. We define linear forms Li on C m and Lj* on Cr by m

r

Li(z) = X aikzk> Lj*(w) = X a kj w k k=l

(z = (zi,...,z m ), w = (wi,...,w r )).

k=l

We suppose throughout this section that (1)

Re(aij) > 0 for all i and j .

This assumption guarantees that Li(z) and Lj*(w) for z e R + m -{0} and w G R+r-{0} stay in H', because H ' 3 R+H1 and H ' D H ' + H ' . Then we formally define the Shintani L-function by (2)

C(s,A,x,x) = X n G N rX n L*(n+x)" s for s = (si

sm) G C m ,

where we write L*(n+x) = (Li*(n+x), ..., L m *(n+x)) G C m and for w = (wi, ..., w m ) G H' m , we write ws = I I ^ w / J . When A is the scalar 1 and % = 1, then

£(s,A,x,%) = ^(s,x) = Z°° (n+x)"s and thus the Shintani n=0

L-function is a direct generalization of the Hurwitz L-function. We leave the proof of the following lemma to the reader as an exercise: Lemma 1. £(s,A,x,%) converges absolutely and uniformly on any compact subset in the region Re(si) > — for all i.

2.4. Shintani L-functions

49

Exercise 1. (a) Prove the above lemma. (Reduce the problem to the case where all entries of A, %, and x are 1 and use the fact that r 1 #{k e Z + | ki + —+k r = n} < Cn*" for a constant C > 0. Actually, £(s,A,x,%) converges if Re(si+---+sm) > r and Re(sj) > 0 for all j.) (b) When all the entries of % are equal to 1 and A is a real matrix, show that £(s,A,x,%) diverges at s = —(1,...,1) (actually it diverges if si+---+sm = r). Now we give another exercise which generalizes the fact that T(s) = P e ' V ^ d y if Re(s) > 0: Jo Exercise 2. If a e Hf and s e H', then

Jot

already explained, a' s = | a r s e r°°

f e ' ^ y ^ d y = a"T(s), where as Jo

writing a = | a | e

-1

ft

__

.7C

with — < a < — . First

1

1

ay

s

y

interpret the integral J e~ y dy as a" times the integral of e" y

on the line

in H' from 0 to +<*> with argument a. Then relate this integral with the T integral J e^y^dy by using the following integral path: and show that the integral on the inner circle of radius e (resp. the outer circle of radius N) goes to 0 as e -^ 0 (resp. as N - > +°o). N

Now we want an integral expression of £(s,A,x,%) converging in the domain with sufficiently large real part. We consider the following function G(y) = G(y,A,x,%) with variable y in R+m given by c

G(y) = X n e N r Xnexp(- ^

(3)

L*j(n+x)Vj).

The convergence of this series can be shown as follows. First of all, we see that

This shows that G(y) =

50

2: L-functions and Eisenstein series

Since Li(y) e H' as already remarked, | %iexp(-Li(y)) | < 1 and the geometric series in the inside summation converges absolutely. We then have (4) Now writing

V( X V G(y,A,x,X) = Uft r1-Xiexp(-Li(y)) ' ff r\ ' y s = n™ = 1 y; s i

for

s e C m , 1 = (1,1,...,1) e

Cra,

dy = dyidy2"-dy m and F m (s) = n^jIXsi), we have, by Exercise 2,

(5) J J - J ~ G(y,A,x,z)y s - 1 dy = JJ...J" In€Nr%nexp(-I™1L*j(n+x)yj)ys-1dy

-s = rm(s)C(s,A,x,x). As in the case of the Riemann zeta function (Proposition 2.1), we can justify the interchange of the integral [)•••[

and the summation £

r

marked by "?" if

Re(si) > — for all i, thus (5) is valid (if Re(sO > — for all i). In fact, if A is a real matrix and % = 1, the convergence of E

N rexp(-Z^ 1 L*j(n+x)yj)

to

r

G(y,A,x,l) is monotone on R+ and hence we can interchange the integral and the summation at the equality marked by "?". We also know from this that G(y,A,x,l)y Re(s) " 1 is an integrable function if Re(sO > — for all i. In general, we know that I G(y,A,x,%)ys"11 < G(y,Re(A),x,l)y Re(s) " 1 and using the dominated convergence theorem of Lebesgue, If a sequence of continuous (actually integrable) functions fn(x) (on an interval [a,b] in R; a and b can be ±oo) is dominated by a continuous and integrable function and f(x) = lim fn(x) at every point x, then n—>o<>

rb

rb

lim fn(x)dx = lim Ja. n—>°°

fn(x)dx,

n—>ooJa

we can justify the interchange. Things have worked in exactly the same way so far, but we encounter a serious difficulty in converting the above integral into the contour integral convergent for all s G C m . Probably, many people before Shintani considered the zeta function of type (2) and tried to get its analytic continuation, but because of this difficulty, we had to wait until 1976 [Stl] to get the analytic continuation of £(s,A,x,%).

2.4. Shintani L-functions

51

First, we explain why the naive conversion to the contour integral does not work. For simplicity, we only treat the case where % = 1 and x = 1. As already seen, words,

e -z

1-e

— has a simple pole at z = 0 whose residue is equal to 1; in other

1-e

—£ is holomorphic at z = 0. Thus each factor of

has a simple pole at the hyperplane S[= {ye C m | Lj(y) = 0} if m > 2 (if m = 1, then Si = {0}). On the other hand, if we denote D(e) = { y e Cl | y | < e } , then D(e) m gives a neighborhood of 0 in C m and thus D(e) m riSi •*• 0 if m > 2. This implies that we cannot avert the singularity by taking the path 3D(e)m. Shintani's idea is to convert the integral (5) into a contour integral by means of an ingenious variable change. We divide R + m into the following m regions: R+ m = U £ = 1 D k ,

D k = {y = (yi, ..., y m ) | y k > yi for all i * k } .

This decomposition can be illustrated in the case of m = 2 as follows: Y2 On each D k , we shall make the following variable change:

(6) (0

) = M(f;,f2,...,rm) k) and tk = 1).


Yi

Thus yi = u t i , y 2 = u t 2 , ..., y k = u, ..., ym = utm. Since the computation is all the same for any of the D k , we assume k = 1 and compute the jacobian matrix:

du

du

a

h

0 0

u 0

0 u

i0

0

0

Thus we know that dy = u m " x dudt and Tr(s) = S1+S2+—f-sm. On the other hand, we see that (7) Thus we have

and

k-l

= uTl(s)'mts-1

for

52

2: L-functions and Eisenstein series m

JJ-J G(y,A,x,x)ys-1dy = £ k=l

= X O o - J o Gk(u)t,A,x)%)uT^-1ts-1dtdu) kl k=l

where G k (u,t,A,x,%) = U],—^—•*—^. This integral expression can be '"'lXexpCuLW) converted into a contour integral convergent everywhere. First let us examine the singularities, i.e., the zeros of

l-% i exp(-uL i (t)). Write %. = l % J e ' i with

0 < 6i < 2K. Then writing z = uLi(t), we see that the zeros of l-%iexp(-z) are located as follows (note that / o g l x j ^ 0 because |%.| < 1): e^ The small circles in the figure at left indicate zeros. Therefore we can take 8 > 0 so that if 0 < | uLi(t) I < 5 for all i. We fix such a 8. Then we can find 0 < 8' so that if | u | < 8' and T

7^-2^

| ti | < 1 for all i, then

| uLj(t) | < 8 . We also

fix such a 8'. On the other hand, we know from (7) that

limRe(Li(t)) = R e ( a i k ) > 0.

Thus we can find

Put

a = Min{Re(a i k ) | i = 1,..., r } .

1 > 8" > 0 so that if

I til < 8" for all i * k, then

Re(Li(t)) > y- Thus the only possible zeros of U\=l{ l-%iexp(-uLi(t))} in the neighborhood U = {(u,ti, ..., t k _i,t k + i, ..., t m ) | | u | < 8', | til < 8M for i * k } are at Ufl{u = 0 } . On the other hand, if R s u > 5 ' and | ti | < 8 " , Re(uLi(t)) > a8'/2 (i.e. | %iexp(-uLj(t)) | < 1), and hence no poles are expected in this remaining case. Thus on the following integral path, we do not have any singularity of G k (u,t,A,x,%) for 0 < e < min(8',8 M ) independent of t and u: P(e):

I

^

P(£,l): U , -

+oo for u

1 for ti,

where the circle is of radius e centered at 0. Note that if t is on the real line, Re(Li(t)) > a always because ay e H' for all i and j . Thus I Gk(u,t,A,x,%) | decreases exponentially as u goes to infinity when t e P(e,l) m " 1 . Thus the integral of u on the real line from e to -h» always converges. On the other hand P(e,l) m " 1 is compact and therefore the integral on this path also converges always. Thus

2.4. Shintani L-functions

53

gives an analytic function of m variables on the whole complex space C m . Thus we have the following result: Theorem 1. £(s,A,x,%) can be continued to the whole space C m as a meromorphic function and has the following integral expression valid for all s e C m :

' "

X )

"

Here we insert a general formula. We assume that %i * 1 for all i. Then G(y,A,x,%) has no singularity on a neighborhood {y | lyil <£ for all i} for sufficiently small e and therefore has a power series expansion in y around 0. We look at the expansion (8a) G(y,A,x,x) -

IT

ex ex P(- x i L i(y))

P

for G(y,A,x,x) in (4), where (n+1)! = n ^ i i i + l ) ! and y" = II™ 1 y i " i . We also write the coefficient of u ^ ' ^ I W i ' " 1 in Gk(u,t,A,x,%) as

B/

^

f

or

/ G N. If %i 9t 1 for all i, then G(y,A,x,%) is holomorphic at y = 0 and Gk(u,t,A,x,%) is holomorphic at u = 0 and t = 0. Thus we can compute the expansion of Gk(u,t,A,x,%) using that of G(y,A,x,%), simply replacing yi by uti (i ^ k) and yt by u. Then, we have (8b)

B n l (x) = m - ^ B ^ C x ) for 1 = (1

1).

Thus we conclude, if Xi * 1 f° r all U that (8c)

C((l-n)l,A,x,x) = (_i)m(n-i)BniM

for

all

0

< n e Z.

This formula will be used later to get a p-adic interpolation of this type of L-function and can be proven without using the variable change y h-> (u,t). Exercise 3. When A is a 2x2 matrix

fa b^

and % = (1,1), get the explicit

formula of £((l-n,l-n),A,x,%) for positive integers n in terms of Bernoulli polynomials given in (3.3b).

54

2: L-functions and Eisenstein series

§2.5. L-functions of real quadratic fields and Eisenstein series In this section, we interpret the Shintani zeta function in terms of Dedekind zeta functions and Hecke L-functions of quadratic fields and later introduce Eisenstein series in this context. Although we are ready to do the same thing for general fields, the detailed exposition in the case of quadratic fields helps demonstrate what is going on. First we treat the quadratic field F = Q(Vd) for a square-free integer d. We take the standard basis {0)1,0)2} of the integer ring O of F ([N-Z, 9.5]); i.e., [Vd if d = 2 or 3 mod 4, 0)1 = 1 and 0)9 = i /— I (l+Vd)/2 if d SE 1 mod 4. Let / be the ideal group of F and I(m) the subgroup of / consisting of ideals prime to a given integral ideal m. Similarly, let ?P+(m) be the subgroup of / made of principal ideals aO with a = 1 mod m and a ° > 0 for all real embeddings a of F into R. Then as seen in Theorem 1.2.1 (see also Exercise 1.2.1), the (strict) ray class group Cl(m) = I(m)/@+(m) is finite. We consider a character % : Cl(m) —> C x . Such a character is called a (finite order) Hecke character modulo m. Then the Hecke L-function of % is defined by £(s,%) = £ « e I(m), oz>n X(n)N(nys

(s e C).

Exercise 1. (a) Show that L(s,%) is absolutely convergent if Re(s)>l. (b) Show that L(s,%) has the following Euler product expansion:

£(s,x) = nPa-x(p)N(Pysy\ where p runs through all prime ideals prime to m. Now suppose d > 0 . Then F = Q(Vd) is a real quadratic field and there are two real places. Let a be the non-trivial field automorphism of F, so that (Vd)a = -Vd. Now O* = {±1 }x{e o n I n e Z} for a fundamental unit eo by Dirichlet's theorem (Theoreml.2.3). Let E = {5 e Ox \ 8 > 0, 5 a > 0}, which is a subgroup of finite index of Ox (in fact E D (O X ) and hence n (O*:E)|4). Then we can find a generator e of E and E = {e |n e Z}. We may assume that e < 1 < e a . The following fact is very important to express the Hecke L-function as a sum of Shintani L-functions. Lemma 1. Let F+ = {a e F | a > 0 and oca > 0}. Then each a e F + c a n b e u n i q u e l y w r i t t e n a s a = e n ( r + s e ) f o r 0 < r e Q , 0 < s e Q and n EZ .

2.5. L-functions of real quadratic fields and Eisenstein series

55

Proof. We embed F into R 2 by X H (x,xa) e R 2 . It is clear from the figure that F R = FQR = R-span of {l,e} = R2, where e = (e,ea) and 1 = (1,1). The curve in the figure is defined by xy = 1. Multiplication by e (as a map of FR into itself) takes the line passing through 0 and 1 to that passing through 0 and e. We denote by V the cone {s«e+r«l I r > 0, s > 0}. Then we see easily from the figure that (R+)2 = U n e Z e n V which is a disjoint union. Thus any a e F+ falls in a unique e n V, namely a = e n (s«e+r-l) for unique r > 0 and s > 0. Since 1 and e form a basis of F over Q, we can find r' and s1 in Q in F. This in particular means that such that e~noc = s'e+r' n e" oc = s'«e+r'«l = s«e+r-l. Since 1 and e form a basis of R 2 , we now know that s = s1 e Q and r = r' e Q. This finishes the proof of the lemma. Now let x : C\(m) -» C x L-function

be a Hecke character and consider the Hecke

£(S,X) = £»<= /(m), oz>« %(n)N(ny\ Let h = #Cl(O) and let us fix a representative set [a\9...,%} for C1(O) consisting of integral ideals (i.e. ideals of O). Then for any fractional ideal 5, we can find a unique a\ such that 6 = a q for a e F + . Applying this to 6m l9 we can find (3 e F+ such that 5m~l = paj, namely, 6 = $a}m. Thus {aim,...,%/n} forms another representative set. Exercise 2. Show that we can take the representative set {#!,...,%} so that each O{ is prime to the given ideal m. Hereafter we extend % to the whole ideal group / so that %(6) = 0 for 6e I-I(m). Then we can write

Therefore to express the Hecke L-function as a finite sum of Shintani L-functions, it is sufficient to do so for the partial L-function

Let Ri = {a e a(lml

| a = xi + x 2 e, 0 < xi < 1 and 0 < x 2 < 1}.

56

2: L-functions and Eisenstein series

This is a finite set. In fact, W = {[0,l]+[0,l]e} is a compact subset of R . On the other hand, a(xml - aZ+bZ fora Z-basis {a,b} since a\Xml is a fractional ideal. Since {a,b} forms a basis of R 2 via the embedding F —» R 2 which takes a h-> ( a , a a ) , a(lml is a discrete submodule of R2 . Thus the subset is discrete and compact Wf)ailml and hence is finite. Then Ri is finite since W z> Rj. T h e shadowed area is W and the dots indicate points in Ri. Now for any Diagram of W and Rt: a £ ^-i ^ i ^ by L e m m a ^ we can uniquely write a = e n (r+se) with 0 < r e Q and 0 < s e Q. Let [s] denote the largest integer not exceeding s and (s) = s-[s]. Thus 0 < (s) < 1. Similarly we define [ if 0 < (r) < 1, and {r} = r-(r)e Z. U if (r) = 0, Then r+se = (r)+(s)e + {r}+[s]e and (r)+(s)e Since Oz> o[m, we have a\ m => O. This shows especially that {r}+[s]e e a( m and hence (r)+(s)e G (0,l]+[0,l)erifli"1w"1 = Ri. Thus, we see that +

= U keZ e k II Xi+X2eGRi {xi+x 2 e+m+ne | (m,n) e N 2 }.

This shows that

where N(xi+x2e+m+n£) = (xi+x2£+m+ne)(xi+x2e a +m+ne a ). Now we shall show that

Exercise 3. Show that there exist y e O and an integral ideal 6 prime to m such that yO = mB. We can write (xi+x2e+m+ne)aim = y(xi+X2e+m+ne)y"1fli^« Since yO=mS, yla\m = OiB'1, which is prime to m, and hence %{y~la\m)* 0. On the other hand, y(xi+X2£+m+ne) = y(xi+x2£)+Y(m+ne) = y(x!+x2£) mod m because

2.5. L-functions of real quadratic fields and Eisenstein series

57

y(m+ne) e yO = mb. Thus %(y(xi+X2E)) = %(y(xi+x2e+m+ne)). (Actually we should remark the fact that y(xi+X2e+m+ne)/y(xi+X2e) € F + to assure this equality). This shows that Thus finally we know that

This combined with the analytic continuation of the Shintani zeta function yields Theorem 1. Let ¥ be a real quadratic field and % : Cl(m) —> C x be a Hecke character. Then the Hecke L-function L(s,%) can be continued to a meromorphic function on the whole s-plane. Moreover, it has the following expression in terms of Shintani zeta functions: eO J,(Xl,X 2 ),l),

where 1 = (1,1) and e is a totally positive fundamental unit of F. The analytic continuation of L(s,%) was first shown by Hecke in 1917. Actually, one can show, by Hecke's method, that L(s,%) is entire if % is non-trivial and only has a simple pole at s = 1 even when % is trivial. Here the word "entire" means that the function is analytic everywhere on the s-plane. We will come back to this question later in Chapter 8. Corollary l(Siegel-Klingen). For a positive integer n, L(l-n,%) e Q(%). Proof. We here give a proof due to Shintani. We will come back later to this problem and give a proof due to Siegel and another proof due to Shimura (see Corollary 5.2.2 and Theorem 5.2.2). What we need to prove is that for a positive integer n, £((l-n,l-n),

o

,(xi,X2),l) e Q.

By the study of the Shintani L-function, we know that

(e 4 *<M)(e 2 ™-l)r(s) 2 | s=1 . n }

58

2: L-functions and Eisenstein series

where

exK-x.ud + t)) x ex P (-x 2 u(e + e«t)) 1 - exp(-u(l +1)) 1 - exp(-u(e + eat)) expC-x^Ct + l)) x exp(-x2u(et + £a)) 1 - exp(-u(t + 1)) 1 -exp(-u(et + e a ))'

0(Uft) =

G a

, t) ^ '

=

Writing (n!)"2B'n(xi,X2) for the coefficient of u^11'1^11"1 in the power series expansion of G(u,t), we see that B'n(xi,X2) is a polynomial with coefficients in F. Moreover if we denote by B'na(xi,X2) the polynomial obtained by applying a to all coefficients of B'n(xi,X2), then (n!)~2B'na(xi,X2) gives the coefficient of U2(n-i)tn-i i n G a (u,t). Thus by the Cauchy integral formula and the fact that

(e4*'s-l)(e2*/s-l)r(s)21 s=1 _ = 2 ? ^ ?

}

,

we have, noting the fact that

(n-1)! Xi

e Q, ^

^ x ^ U )

= r 1 n- 2 Tr F /Q(B l n (xi > x 2 )) e Q.

We now want to introduce Eisenstein series which will be studied in detail in Chapters 5 and 9 and which is one of the simplest examples of modular forms. First let us give a brief definition of modular forms. We write H = {z € C I Im(z) > 0} for the upper half complex plane. Then the group Gl4(R) = {oce M 2 ( R ) | d e t ( a ) > 0} acts on H via zh-> cc(z) = ^ ± 1 cz + d for a = , (cz+d * 0 because z is not real). We see easily that (a(3)(z) = a(p(z)). To see a(z) stays in H, we use the following identity: for z1 = oc(z), a bYz w\ fa(z) a(w)Ycz + d 0

c dj[l

lj

=

[ 1

1 A 0

cw + d

Then replacing w by the complex conjugate of z and taking the determinant, we see that det(oc)Im(z) = Im(a(z)) |cz+d| 2 and hence if det(oc) > 0 and Z G H , then a(z) e H. For any discrete subgroup T of GL^R), a modular form on T of weight (s,k) (s e C and k e Z) is a function f: H -> C such that f(Y(z))=j'(y,z) k lj(y,z)| 2s f(z) for all ye T, where j(y,z) = det(y)"1/2(cz+d) if y =

, . We see easily that

j(y8,z) = j(y,8(z))j(8,z) (a cocycle relation). Then we see that f(yS(z)) = j(y5,z)k | j(y5,z) 12sf(z) = j(y,8(z))k | j(y,8(z)) 12sj'(8,z)k | j(8,z) 12sf(z) = J(T,5(z))k | j(y,8(z)) 12sf(8(z)) = f(y(8(z)))

2.5. L-functions of real quadratic fields and Eisenstein series

59

and hence the definition is at least consistent. When f is holomorphic (resp. C°°-class), then f is called a holomorphic (resp. C°°-class) modular form. The importance of modular forms lies in the fact that it is a non-abelian replacement of Dirichlet and Hecke characters (see Chapter 9 for more details about this fact). This point will be clarified later in Chapters 8 and 9. We will study modular forms in detail in later chapters: Chapters 5 to 10. Here we introduce an example of modular forms: the Eisenstein series. Take a positive integer N, integers 0 < a < N, 0 < b < N (so that (a,b) * 0), and k > 0. Put as a function of z e H and s e C

I mZ+n I

E'k,N(z,S;a,b) = £ ( m , n ) G Z 2. { 0 } , (m,n).(a,b) mod N

One can easily verify that this series is absolutely and locally uniformly (with respect to s and z) convergent if Re(s) > l-y- We can write E' k ,N(z,S;a,b) = 2,(m,n)eZ2-{0}, (m,n)E(a,b)modN(mZ+

mz+n

I -2s

= X'(m,n)Ez2((a+Nm)z+(b+Nn))-k I (a+Nm)z+(b+Nn)

-2s -2s

where

Z'

means the summation over all

(m,n) e Z

for which

(Tr+m)z+(TT+n) * 0. To relate this series with Shintani L-functions, we consider, for 0 < u < 1 and 0 < v < 1, q>k(z,s;u,v) = X f (mn)eZ 2 ( u z + v + m z + n )~ k ' uz+v+mz+n | "2s. Then we have -k_ /_ _ a b x E' k , N (z,s;a,b) = (1) We now split the summation of 9k into four pieces:

Here we write u* = 1-u (0 < u* < 1) and v* = 1-v (0 < v* < 1). We consider the function summed on a cone: £(CDI,G)2;S;U,V) = X(m,n)€N2(ucoi+vco2+mcoi+no)2)"k I ucoi+vco2+mcoi+nco21

-2s

60

2: L-functions and Eisenstein series

for a>i,G)2e C and for u, v not both 0- Then it is easy to see from the above figure that, except when (u,v) = (0,1), q>k(z,s;u,v) = 5(zJ;s;u,v)+^ When (u,v) = (0,1), we have 9 k (z,s;0,l) = §(z,l;s;(U) + $(-z,l;s;l,0) + £(z,-l;s;l,0) + £(-z,-l;s;0,l). We can choose a,p G C X SO that cc,ocz e H' and P,Pz G H' for any given z E C -R. The choice of p = a works well, but we keep the freedom of choosing P differently. Then we see that uocz+va+naz+ma G Hf and upz+vp+npz+mp e H' for any (m,n) e N 2 . Recall that (xy)s = x s y s if x,y e H', where we define xs by I x | seI'8s for x = | x | e1"9 with - - < 0 < - . Thus we have (uccz+va+maz+na)'s(uP z +vp+mP z +n P )"s = {(uaz+va+maz+na)(uP z +vP+mp z +np)} "s = {aP | uz+v+mz+n 12}"s = (ap)" s | uz+v+mz+n | ~2s. From this, we conclude that

2(uaz+va+maz+na)" k " s (up z +vp+mp z +np)' faz )^ In the above argument, replacing (u,v) by (u*,v*), (z,l) by (-z,-l) and (a,p) by (-a,~P), we obtain faz FQz\

(-a)-k(ap)-s^(-z,-l;s;u*,v*) = C((k+s,s),^a

p

J,(u*,v*),l).

Similarly choosing a',p' e C x so that a'(-z),a' e H' and p'(-z),p f G H \ we have

and

(-aT T kk(oc'PT ( o c 's^z,-l;s;u,v*) P T s ^ l * ) = C((k+s,s),^, C ( ( k + ) ^ ZZ Thus we have a theorem of Shintani ([St6, §4]):

"p(ZZ))J,(u,v*),l).

2.5. L-functions of real quadratic fields and Eisenstein series

61

Theorem 2. The Eisenstein series E'k,N(z,s;a,b) can be continued to a meromorphic function of s to the whole plane and is real analytic with respect to z. Moreover writing u = (-^, ^ ) and supposing u * (0,N) ifN > 1 and u = (0,1) if N = 1, we have the following expression in terms of the Shintani L-function:

E'k,N(z,s;a)b) = N- 2 s - k {(ap) s a k C((k+)H

j

(a'p')salkC((k+s,s),r'

J,u*,l) faz

a,|3,a',p' are m/:en ^ r/zaf a z , a e Hf, Pz,P e H1, a'(-z), a ' € H' p'(-z),p'e H \

and

The analytic continuation of Eisenstein series was first proven by Hecke in the 1920's. Exercise 4. (a) Show that the following subset of SL2(Z) is a subgroup of finite index:

T(N) = {a e SL 2 (Z) | a = L

A mod NM 2 (Z)}.

(b) Show that E'k,N(z,s;a,b) is a modular form of weight (k,s) on the subgroup

r(N). Now we compute the residue of E'o,N(z»s;a,b) at s = 1, which is a key to proving the class number formula for imaginary quadratic fields. We first deal with the residue at s = 1 of the Shintani zeta function. For each matrix A =

(a. b {c d

with a,b,c,d € Hf, we have seen the following integral expression: (e 47lfs -l)(e 27lIS -l)r(s) 2 C((s,s),A,x,(l,l)) = Jp (e )Jp ( e,i) G i( u ' t )u 2s - 1 t s - 1 dtdu + J p(£) Jp (el) G 2 (u,t)u 2s - 1 t s - 1 dtdu, where Ullu tj

'

_ " _ -

fexp(-xiu(a+bt))l [l-exp(-u(a+bt))j fexp(-xiu(at+b))l j j

x

x

Jexp(-x2u(c+dt))] {l-exp(-u(c+dt))f Jexp(-x2u(ct+d))] {l-exp(-u(ct+d))J*

Even if t moves around the interval [0,1], Gj(u,t)u is finite provided that (Gj(u,t) has a pole of order 2 at u = 0). Thus if Re(s) > 0, we do not have to make the contour integral with respect to t; that is, we have if Re(s) > 0

62

2: L-functions and Eisenstein series

(e 47l/s -l)r(s) 2 C((s,s),A,x,(l,l)) G2(u,t)u2s-1ts-1dtdu. Note that the expansion of Gi(u,t) at u = t = 0 is, for the Bernoulli polynomial Bj(x) introduced in §3, ix v(u(c+dt))k-1 2/j=o v-lrBjCxi) j, 2/k=o ^ Bk(X2) k! In particular, the coefficient of u"2 is (a+bfT^c+dt)"1. Now we compute the value 0

1 dt ( a + b t ) ( c + dt)

A

A.tl{A) {-log(2i+b)+log{c+d)+log{&)-log{c)} if det(A) * 0, ^

. f d . . ( A , - 0 .

We can compute similarly the integral: J p(£) J

G2(u,t)u2s~1ts~1dtdu and have

Jp(e)Jo G 2 (u,t)u 2s - 1 t s - 1 dtdu f

1

d

)

-

Z

o

^

(

d

)

}

if det(A) / 0,

. f 4 . . ( A , - 0 . Exercise 5. Explain how one can compute J

. . y ^^dt.

Now we start the computation of the residue of E'ojvjCz.sja.b). We have (<xz

i(a'p')sC((s,s)^ a, =

p

J.Cx.x1),!)

-2-i(z-zy1{log(-a'z)-log(-pz)+log(p)-hg(a')}.

If ze IT, then z € H', and a and (3 can be taken in H1. Hence log(az) = log(a)+log(z) and log($z) = log($)+log(z). On the other hand, if we write a 1 = I a' I e m and z = | z | e10, then — < 0 < — 2 2 and — < 0 —7i + a<— a n d — < a < — . This shows 2 2 2 2 log(-a'z) = log{ | oc'z | )+(a+0-7c)/ = Similarly, if p1 = | pf | c^, then - | < %-Q+b < \ shows that

and - | < b < \

This

2.6. L-functions of imaginary quadratic fields

63

(-$ z) = log( I p'z | Thus we have ccz

Thus Ress==iElo N(z,s;a,b) = —9 . One can easily verify the same formula Nlm(z) even if Re(z) < 0 and we can conclude with Corollary 2.

Ress=iE'0,N(z,s;a,b) =

2

Exercise 6. Prove the above formula when Re(z) < 0.

§2.6. L-functions of imaginary quadratic fields In this section, we interpret special values of Eisenstein series as the values of L-functions of imaginary quadratic fields and later prove the class number formula. Let F = Q ( V - D ) be the imaginary quadratic field with discriminant -D and m be an integral ideal of O. We consider a quasi-character % : I(m) —> C x such that %((a)) = a k a c m for a = 1 mod m, where c denotes complex conjugation, occm = (a c ) m , (a) = aO is the principal ideal generated by a and (k,m) is a pair of integers. Such a character is called a Hecke character (of ©o-type (k,m)). When (k,m) = (0,0), % gives a character of the ideal class group Cl(m) = I(m)/2(m). The Hecke L-function is then defined by

^X)

= yLn.I(m),o^X(n)N(nr

(s E C).

Note that %(ot) = a k oc c m = a k - m N ( a ) m = a c ( m " k ) N ( a ) k . Thus writing N : 1(0) —» C x for the Hecke character given by N(a) = NF/QU) for all ideals a, L(s,%) =L(s-m,Xi) = L(s-k,%2) for %i = %N'm and %2 = XN'k. Note that %i is of type (k-m,0) and yji is of type (0,m-k). Thus without loss of generality, we may assume that % is of type (k,0) or (0,k) for k > 0. Since the argument is the same in either case, hereafter we assume that % is of type (0,k) with k > 0. Exercise 1. Assume F = Q (V d) for a square-free integer d > 0. Let % : I(m) -> C x be a quasi-character such that %((cc)) = cc k a a m whenever a = 1 mod m for a pair of integers (k,m) and the unique non-trivial field automorphism a of F. Show that k = m. (Use the existence of non-trivial units.)

64

2: L-functions and Eisenstein series

By Exercise 1, we do not lose generality by assuming k = m = 0, i.e. % is of finite order, when F is real quadratic. As in the case of real quadratic fields F, we take a representative set {a\,...,%} of Cl(m) consisting of integral ideals. Then we see that

where [i(m) = {£ E ( f l ^ s 1 mod /rc}, which is a finite group and trivial if m is sufficiently small. To express S aGa .-i a H l moda -i m a ck A^(a)" s as a sum of special values of Eisenstein series, we pick a basis (0)1,0)2) = (0)14,0)2,1) of of1 and a positive integer N such that N e m. We may assume that a* is prime to N and Z{ = 0)1/0)2. By changing 0)2 to -0)2 if necessary, we may assume that Zi = 0)1/0)2 e H. Let R i = {(a,b) e Z 2 | 0 < a < N, 0 < b < N and ao)i+bco2 = 1 mod

a(lm\.

Then Ri is a finite set, and we have

ckx

(ao)i+bo)2+Nmo)i+Nno)2)"k (a,b)ERi(m,n)EZ2 I (ao)i+bco2+Nmo)i+Nno)2) 12s"2k k = £ (afb)eRi I ^ I -2s+2kE'k,N(0)i/0)2,s-k;a,b). ( a f b ) e RG>2iG>2 Thus we have

/(m), oz>nX(n)N(nys I ^ ( w ) I ' 1 ^(a,b)ER i c °2,i" k i co2,i I • 2 s + 2 k E l k f N (z i ,s-k;a > b). Theorem 1. L(s,%) ca« Z?e continued to a meromorphic function on the whole s-plane. In particular, when k = 0, and m= O, and % is trivial, then CF(S) = w-1X"1^(«i)"Ilo)2.il"2lEIo.i(zi.s;O,l), where H is the class number of F and w = \(f\. 1 2

Idetf" "

Note that

2.6. L-functions of imaginary quadratic fields

65

On the other hand, we see that fCGj det _

©2^ _ _ _ , 9 _ = O i CG2-CO2CO1 = 0)2 0)2(zi-Zi) = l o ) 2 r 2 V - H m ( z i ) .

By the residue formula of E'o,i(zi,s;O,l) (Corollary5.2):

we have Dirichlet's residue formula: Theorem 2. Let H be the class number of the imaginary quadratic field of discriminant -D. Then we have —T=.

wVD

Let R = Z [V^D] 3 Z. Thus R is a subring of the integer ring O and R = Z[X]/(X 2 +D). Let p be an odd prime in Z prime to D. Then we see easily that X +D mod p is reducible if and only if -D is a quadratic residue mod p.

In fact, if

—

= 1, then we can find

a e Z

such that

a 2 = -D mod p and R/pR = F[X]/(X-cc)(X+a) = F 0 F for F = Z/pZ. Thus pR = p\C\p2 m R fo r two distinct prime ideals p\ and p2> If D = 0 mod 2, then O = R and thus (1) If

pO = p\p2 for p\*p2 D

is odd, then for

co =

if and only if ,

—

= 1.

the minimal polynomial of

2

co: X -X+N(co) = 0 is reducible over F if and only if there is a e Z such that a 2 =-D mod p, because 2 is invertible in F. Thus (1) is still true. For example, if D is a prime, then D = 3 mod 4 (i.e. (=£\ = (-l){D'1)/2

= -l),

and we know from the quadratic reciprocity law that

D

Thus the map p H-> —

is a Dirichlet character modulo D. More generally, we

have the following fact: Exercise

2.

Using the quadratic reciprocity law, show that the

map: p n

—

is induced by a Dirichlet character %D : (Z/DZ)X -> {±1}.

Thus we see that pO = p\p2 if and only if %D(P) = 1- Now we look at the Euler factor of the Dedekind zeta function of F. We see

2: L-functions and Eisenstein series

66

XD(P) = ! XD(P) = - 1 XD(P)

=0

Euler factor at prime ideals dividing p (l-iV( Pl )- s )(l-N(^)- s ) = (l-p"s)(l-XD(p)p-s) l-N(pYs = (l+p- s )(l-p- s ) = (l-p- s )d-% D (p)p- s ) l-iV(p)-s = (l-p- s )(l-x D (p)p- s )

Thus we have CF(S) = Up Thus we know that

l

= C(s)L(s,%D).

L(l,x D ) = Ress=iC(s)L(s,%D) = Res s= iC F (s) = On the other hand, by the functional equation, we can relate value L(0,%D) = -D"1I^:i1xD(a)a (see §3). Thus we have

L(1,%D)

with the

Theorem 3 (Dirichlet's class number formula). H = -—X a =Ti %D(a)aExercise 3. Using the above class number formula, show (i) For a prime p > 3 with p = 3 mod 4, the number a of quadratic residues in [0, S] exceeds the number b of quadratic non-residues in the same interval; (ii) If p > 3 and p = 3 mod 8, then a-b = 0 mod 3.

§2.7. Hecke L-functions of number fields Let F be a general number field and let I be the set of all embeddings of F into C. Let I(R) be the subset of I consisting of real embeddings and put I(C) = I-I(R). Then the number of real places of F is given by r = #I(R) and the number of complex places is given by t = #I(C)/2. We start with the study of the fundamental domain of F+/E, where E = ( e e O x | £ ° > 0 for all o e I(R)}, F+ = {a e F x I a a > 0 for all a e I(R)}. Thus if I(R) = 0 , then we simply put F + = F x . The result we want to prove first is Theorem 1 (Shintani [Stl, St5]). Let E' be a subgroup of finite index in E. Then there are finitely many open simplicial cones Q = C(vii,...,Vimi) with vij G F+ such that C = UjCi and F + = U£GE-eC are both disjoint unions. We can take the Q's so that there exists ua>i e C x for each a e I and i such that Re(ua)iVija) > 0 for all j = 1, ..., mi.

2.7. Hecke L-functions of number fields

67

Here an open simplicial cone C(vi, ...,v m ) in an R-vector space or Q-vector space V with generators vi e V is by definition C(vi, ..., v m ) = {xivi+---+x m v m I Xi > 0 for all i}, where the m vectors v^ are supposed to be linearly independent. We divide I(C) = X(C)U£(C)c into a disjoint union of two subsets X(C) and its complex conjugate Z(C)c for complex conjugation c and write £ for I(R)LE(C). In the theorem, we regard Q as an open simplicial cone in the real vector space V = F ^ = F<8)QR = R I ( R ) x C I ( C ) . We then embed F into V by a h-> (oc°)ae £ G V. Then F is a Q-vector-subspace of V which is dense in V. We put V + = R + I < R >x(C x ) L(C) , where R + = ( x e R

| x > 0 ) . Then F + = V+flF.

Proof of Theorem 1. Since the proof is the same for any E', we simply treat only the case E' = E. Consider the hypersurface X in V+ defined by X = {(x a ) a G s | N(x) = IT

aG

i

|2 =

Then, for S = {x e C \ | x | = 1}, we have p : X = S^R1"""1. In fact the projection to S l can be given by x h ) (xa/1 x a | )aeZ(C) G Sl and the projection to R1**"1 is given by xi—> /(x) = (/a(xa))z-{T}» where we exclude one embedding x e Z and / a (x a ) = log{ I x a |) or 2log( I x a |) according as a is real or complex. By definition E acts on X by componentwise multiplication. The image of E in R1"1"*'1 is a lattice by Dirichlet's theorem (Theorem 1.2.3) and hence X/E is a compact set. Thus we can find a compact subset K of X such that X = U e G E eK. We can project V+ to X via x H> N(x)~ 1/d x for d = [F:Q], which will be denoted by n. This is obviously continuous and surjective and hence takes the dense subset F + to a dense subset of X. We can find a small neighborhood U of 1 in the multiplicative group X such that eUflU = 0 if 8 * 1 in E. We may assume that U = CoflX for an open simplicial cone Co in V+ with generators in F+. Thus UXGKfto(F ) x ^° -^ ^* Since K is compact, we can choose finitely many xi e 7i(F+)flK such that UjLjXiCo ^ K. We write F+flxiCo = C0,i. Note that eCo,iflCo,i = 0 if 6 * 1 because eUflU = 0 if 8 * 1 in E and Co is the R+-span of U. Moreover Co,i is a cone with generators in F + . In fact, taking yi e F + such that 7t(yi) = xi, then Co.i = XiCo = yiCo. Since Co = C(vi, ..., Vd) with Vi G F + , we see that Co,i = C(yjVi, ..., yiv^) is a cone generated by vectors in F+. Now admitting the following lemma, we finish the proof of the theorem:

68

2: L-functions and Eisenstein series

Lemma 1. Let C and C be two polyhedral cones whose generators are in F (where a polyhedral cone means a disjoint union offinitely many open simplicial cones). Then CflC, CUC and C-C are all polyhedral cones whose generators are in F. We have F + = U" =1 U eeE eC 0 ,i and Co/leCo,! = 0 if e * l . If n = l , C0,i is the desired cone. When n > 1, we define C i j = Co,i and for i > 2, Ci,i = Co,i-U eeE eCo,i. Since Co,i ( i > l ) intersects with eCo,i for only finitely many e, Cij is a polyhedral cone by the lemma. We now have ( i > 2 ) and F + = U i = 1 U £ G E e C i , i and Ci,in{U£GEeCi,i} = 0 Ci,iDeCifi = 0 if e * 1. We now construct inductively (on j) polyhedral cones Cj,i with generators in F+ for each 0 < j < n by Cy = Cj.^i if i < j and Cjj = Cj_i,i-UeeEeCj_ij (Cy = CJ.IJ) for j < i. Then we see that Cj,inU £GE eCj, k = 0

for i > j > k, F + = Ur=1UeEEeCj,i,

and Cj/leCj.i = 0 if e * 1. Then Cn,i is a disjoint union of finitely many simplicial cones which give the desired simplicial cones. This proves the first assertion of the theorem. We can subdivide the cones Cj so that the last assertion follows. Exercise 1. Give a detailed proof of the last assertion of Theorem 1. Proof of Lemma 1. We may assume that C and C are open simplicial cones. Write C = C(vi, ..., v m ), where the vi's are linearly independent. By adding v m + i, ..., v 0 for 1 < i < m and vi*(v) = 0 for i > m } . Since the vi are in F, vi* induces Q-linear forms on F. (We call such a form a Q-linear form.) Similarly, there are d linearly independent Q-linear forms Wi* on F and an integer n between 1 and d such that C = {v e V | wi*(v) > 0 for l < i < n

and wi*(v) = 0 for i > n } .

Thus C-C and CflC are disjoint unions of sets of the following form for finitely many Q-linear forms Li on a vector subspace W of V (W may not be a proper subspace but W is an R-span of a Q-vector subspace of F): X = {w

G

W | Li(w) > 0 for i = 1, ..., / }.

2.7. Hecke L-functions of number fields

69

We may assume that {Li} is a minimum set of linear forms to define X. For example, when X = CDC, then we take W to be ( x e v | Vi*(x) = 0 and Wj*(x) = 0 if i > m and j > n } . Since CUC = (CnC^LKC-C'jUCC-C) is a disjoint union, it is sufficient to show that X is a polyhedral cone with generators in F. When dim(W) = 1 or 2, the assertion is obvious. Thus we shall prove the assertion by induction on dim(W). Let Xj = {w G W | Li(w) > 0 for i * j and Lj(w) = 0}-{0}, X = {w e W | Li(w) > 0 for i = 1, ..., /}-{0}. Since Xj is contained in Ker(Lj) which has dimension less than dim(W), by the induction hypothesis, Xj is a disjoint union of open simplicial cone. Moreover, Xj-Ui^jXi is a disjoint union of open simplicial cones, and hence it is easy to see that X-X = UjXj is also a disjoint union of open simplicial cones. Write these cones as UjXj = UkC(vki,...,Vkik). Let u be a point in XflF, which exists because X is open in W and Ff|W is dense as already remarked. Since v ki>---»vkik a r e i n a proper subspace of W and u is not in the subspace spanned by Vki,...,Vkik, u,Vki,...,Vkik are linearly independent. Write Ck(u) for C(u,Vki,..-,Vkik). Then we claim that X = UkCk(u)UR+u (disjoint). By definition, Li(x) > 0 for all i if x e X. Thus in particular, if x = Xu for X e R , then Li(x) = A,Li(u) > 0, Li(x) > 0 and Li(u) > 0. Thus, RuflX = R+u. Suppose that x e X is not a scalar multiple of u. Let s be the minimum of Lj(x)/Li(u) for i = 1,..., /. Then s > 0. There is an index i (maybe several) such that s = Li(x)/Li(u). Then Lj(x-su) = Lj(x)-sLj(u) > 0 and Li(x-su) = 0. Thus x-su e X-X. Therefore x-su G Ck for a unique k (because X-X is a disjoint union of the Ck's) and thus x G Ck(u). This shows the desired assertion. Exercise 2. Write down the proof of X-X = UjXj explaining every detail. Let Z[I] be the free module generated by embeddings of F into C. We can think of each element t, e Z[I] as a quasi-character of F x which takes a e F x to a^ = I I a G l a a ^ a G C x . A quasi-character X : I(m) -» C x for an ideal m of O is called an arithmetic Hecke character if there exists ^ G Z[I] such that X((a)) = oc^ for all a G P(W), where P(m) = {a G F+ I (3(a-l) G m for some (3 G O prime to m}.

70

2: L-functions and Eisenstein series

Theorem 2 (Hecke). If X is an arithmetic Hecke character modulo m, then L(s,X) = £ne /(m), ODn X(n)N(n)~s can be continued to a meromorphic function on the whole complex s-plane C. Proof. Let {a\9 ...,%} be a representative set of ideal classes for //fP+ (see Exercise 1.2.1). We may assume that a\ are all prime to nu Since still gives a representative set, we can write

Now we take the fundamental domain C = U ^ C j for F+/E with disjoint open simplicial cones Cj. Here note that for a positive rational number u, Cj = uCj by definition. In particular, NCj = Cj for any positive integer N. Thus we may assume that Cj is generated by totally positive integers in 0. Fix a set of generators {vi,..., vb} of Cj in O and consider the Shintani L-function where v a = ( v i a , ..., v b a ) e C b , x = x 1 v 1 + «"+x b v b with Xi e (0,1], x a = viaxi+---+VbaXb and n«va = vi a ni+--+Vb a nb. By Theorem 1, we also have u a e C x such that all the entries of u a v a have positive real parts. Then

where we have to choose the branch of /
L(S,X) = £ . = 1 XaE(aimnF+)/E

Xt In fact, choosing yeO

J X x ^ C j ( ( s - § a ) a € I , x ) for SG C. such that yO = mB and 5 is prime to m, we can write

(x+p)ai/rt = Y(x+P)y'1Oim. Since yO = mB, yla\m

= a\b"x, which is prime to

2.7. Hecke L-functions of number fields

71

m and hence X(yA 0{m) ^ 0. On the other hand, y(x+p) = yx+yP = yx modx m because yP e yO = mB. Thus ^(y(x+P)/yx) = (y(x+P)/yx)^. (Actually we should remark the fact that y(x+(3)/yx e F + to assure this equality.) This shows that

which finishes the proof of Theorem 2. Exercise 3. Write down every detail of the proof of the equality A,((x+p)aiwi) = (x+p)^x"^(xfljm), because in the proof we implicitly assumed that xo{m is prime to nu

Corollary 1. // n e Z with n < £ a for all a, then L(n,X) e Q(k). This has meaning only when F is totally real, because we will see in Chapter 8 that L(n,X) is zero otherwise. Now we briefly discuss Hilbert modular Eisenstein series. We will take up this topic in adelic language later in Chapter 9 in more detail. Here we suppose that F is totally real, i.e. I = I(R). Consider SL2(0) and its congruence subgroup for each integral ideal m i fl 0>l T(m) = {a e SL 2 (O) | a = I Q x I mod mM2(O)}. We consider the product H1 of d copies of the upper half complex plane H indexed by the set I of embeddings of F into R. Then we let SL2(0) acton H1 fa bY faCT bCT>| so that cc((zo)) = (or(z a )), where ^ d J = | o dCTjE SL 2 (R) acts on each component H via the linear fractional transformation. To define modular forms in a weak sense, we consider not only weights in Z[I] but also those which are formal linear combinations of symbols a and ac with o e l , where c denotes complex conjugation. We denote by Z[lUlc] for Ic = { a c | a e l } the module of weights. A function f i r f - ^ C is called a Hilbert modular form on T(m) of weight (k,s) if f(y(z))=j(y,z) k |j(y,z)| 2s f(z) for all y e where k = S a e n j i c k a a e Z[lLJlc], s e C and for y = j(y,z)kk

0 1

2s

T(m\ fa b c

di

= = noeit^zo+d^^zo+d ) ^} I j(yz) 1 = n aG i I (cazo-fda) | 2 s Then, for (a,b) e (O/m)2, the Eisenstein series of weight (k,s) is defined by

72

2: L-functions and Eisenstein series

E'k>m(z,s;a,b)

where the summation runs over {(m,n)e O2-(0,0) | (m,n) E (a,b) mod and

Here E(m) = ( e e E | e = 1 mod m], which is a subgroup of E of finite index, and E(m) acts on O2 by (m,n)e = (me,ne). Then E'k,m(z,s;a,b) is convergent if s has sufficiently large real part. We now fix finitely many simplicial cones Cj so that C = UiCi gives a representative set of Fx/E(m). Exercise 4. Show that (F x ) 2 = U ee E(m)e(CxFx) is a disjoint union. By dividing F x as a disjoint union of finitely many simplicial cones C j , we know from Exercise 5 that (F x ) 2 = U eGE(m) e{U iJ C i xC l j}. Note that Q x C j is again an open simplicial cone C({(vi,0),(0,Wj)}) for Vi e Q and Wj e Cj. We define, for x = xivi+---+x a v a with xi e (0,1] and y = yiwi+---+ypwp with y i e (0,1], Cij(s,x,y) = X(m,n)ENaxNbnaeI(x°za+ya+m.v°z0+n.wa)"Sc. Then in exactly the same manner as in the proof of Theorem 2, we can express E'k,m(z,s;a,b) as a finite sum of Cij(( s +k a ) ae uji c ,x,y) for s e C. Since £ij(s,x,y) can be continued to a meromorphic function on the whole s-plane and gives a real analytic function of z for a fixed s, we have Theorem 3 (Hecke). The Eisenstein series has a meromorphic continuation to the whole s-plane. When E'k,m(z,s;a,b) is finite at se C,then E'k,m(z,s;a,b) gives a real analytic Hilbert modular form ofweight (k,s) on T(m). We will return to the study of Eisenstein series in more detail later in Chapters 5 and 9.

Chapter 3. p-adic Hecke L-functions In this chapter, we first construct p-adic Dirichlet L-functions for Q using Euler's method of computing the values L(n,%). We follow Katz's article [K2] in this part. Then we generalize the result for Hecke L-functions of totally real fields using the method in [K6]. Roughly speaking, a p-adic L-function is a p-adic analytic function whose values coincide with those of its complex counterpart at enough integer points to guarantee its uniqueness. Here we mean by a p-adic analytic function on an open set U of Z p a function on U having values in a finite extension F of Q p which can be expanded into a power series in (z-u) (with coefficients in F) for each u e U convergent on an open neighborhood of u in U. A p-adic meromorphic function on U is a quotient of two p-adic analytic functions on U. The p-adic L-function we discuss was first constructed for Q by Kubota and Leopoldt [KL] as a continuous function and later shown to be p-adic analytic by Iwasawa [Iwl]. The result was generalized to totally real fields independently by Deligne and Ribet [DR], Cassou-Nogues [CN] and Barsky [Ba]. Katz's method in [K6] is an interpretation of the methods of [CN] and [Ba] in terms of formal groups.

§3.1.

Interpolation series

We recall the definition of the binomial polynomial: as a rational polynomial with variable x, for a non-negative integer n,

©•

lx

n!

1

if

"

*•

x

'

n = 0.

Then it is obvious that this polynomial has integer values on the set N of non-negative integers.

For negative integers

-m

( m e N),

we have

("™) = ( - l ) n ( m + n 1 1 ' 1 ) G Z. Thus in fact, ( * ) has integer value on Z. Such a polynomial is called numerical By the binomial theorem, we have a formal identity in the formal power series ring n=0

When XG N ( = { n e Z | n > 0}), this power series is actually a polynomial of degree x and the identity holds for any value of X. Thus, for x e N,

By comparing the coefficients of X m of (1+X)x(l+X)y = (l+X) x+y , we have

74

3: p-adic Hecke L-functions

i coca

n=0

Let K be a field extension of Q and A be a subring of K. Let f: N -> A be any function. We define the n-th coefficient an(f) of f by

(2)

an = an(f) = X(-l)k(jj)f(n-k) = X(-l) n - k (j)f(k) e A. k=0

k=0

Using these coefficients, we define a new function f* : N -> A by f x = a (f) for XG N =

*( ) XI o * (n)

«

This is actually a finite sum X an(f) ( n ) because ( _ ) = 0 when n > x. We n=0 n=0

V n /

V n /

now compute the value of f*. By definition, we have

( j){t n=0

n=0 [ k = 0

The last equality follows from (la). This shows that

(3)

f(x) = XI = 0 a n®(n) on N.

Exercise 1. Let (ai,..., ar) be an r-tuple of non-negative distinct integers and (bi,..., br) be an r-tuple of real numbers. Show the existence of a polynomial P with coefficients in R such that P(aO = bi for all i = 1, ...,r. We now prove the uniqueness of the expansion (3). We suppose the existence of coefficients {a' n eK} n G N such that Then by (3), putting b n = an-a'n, we see that Supposing the existence of coefficients b n , not all zero, we take the smallest n with b n * 0. By the minimality of n, b m = 0 for all 0 < m < n. Thus

O » G ) - 0 and in particular X ^ b m ^ - O . Since (£)=0 if n < m, we conclude from this that b n = ( n ) b n = 0, which contradicts our

3.2. Interpolation series in p-adic fields

75

choice of n. Thus b n = 0 for all n and the expansion as in (3) is unique. Summarizing this, we have Proposition 1. For a given function

f: N ~> A, there exists a unique se-

quence of elements {an(f)}neN in A such that { ( n ) } n G N 8^ves

x e N. Moreover

a

f(x) = ^ ~ = 0 &n(n) for all

basis of the ring of numerical polyno-

mials.

§3.2. Interpolation series in p-adic fields We are now able to describe, using interpolation series, the space C(Zp;A) of continuous functions on Z p having values in any closed subring A of the p-adic completion ¥p of a number field F. A continuous function f: Z p -» Op is in fact determined by its restriction to the natural numbers N by the density of N in Z p (see (1.3.3)). Thus f is uniquely determined by the interpolation series of its restriction to N: (1)

f(x) = X ^ = o a n (f)(*)

on N.

We are going to show that the right-hand side of the above identity converges in Op uniformly for any x e Z p . After showing this, we will know that the righthand side gives a continuous function which coincides with f on N and hence is equal to f on all Z o . What we need to show is lim an(f) = 0. Then the convergence of (1) and the uniformity of the convergence follow from the strong triangle inequality (1.3.1) and the fact (see (2) below) that | ( * ) | p < 1 for all x € Zp. Exercise 1. Show the convergence of (1) and the uniformity of the convergence assuming lim an(f) = 0. n-»oo

Let us now show that

Urn an(f) = 0.

For any given polynomial

P(X)

d

(= ao+a]X+- • *+adX ) with coefficients in Q p , we see that

I P(x)-P(y) I p < max (| an | p | x-y | p | x^+x^V**'+yn~11 p) 0
< max(|a n | p )|x-y | p on Z p . Thus x h-» P(x) is continuous on Z p . In particular, the function x h-» ( nj

is

continuous on Z p . Since this function has values in N on the dense subset N of Z p , its continuity shows that it takes Z p into Z p . Thus

(2)

ICD'P-1

fora11 XG

Z

P

and nG N

*

76

3: p-adic Hecke L-functions

Since Z p is compact (because it is a projective limit of compact (even finite) sets Z/p n Z [Bour2,1.9.6]), any continuous function f: Z p ~> Op is uniformly continuous [Bour2, II.4.1]; thus, for any given positive e, we can find a small positive 8 such that | x-y I p < 8 => | f(x)-f(y) I p < e. Now assuming x,y e N, we take e to be p"s and write 8 = p~l (t = t(s)) for this e. We may of course assume that t > 0. Then | f(x+pl)-f(x) | p < p' s . If we let fy(x) = f(x+y), then by definition, we have

Using the identity (Lib), ( x + y ) = S t o C X n L ) ' w e h a v e

m=0

n=0

Now we are going to interchange the summation with respect to m and n in the above formula. Since x,y e N, m (resp. n) actually runs from 1 (resp. 0) to x+y (resp. m), the summation is in fact finite and thus the interchange of the two sums is legitimate. Thus we have

n=0

m=n

n=0

k=0

In particular, we have, by the definition in § 1 of a n (f y ),

(3)

2>n + k(f)(k) = an(fy) = XM) n " k (J)f(k+y). k=0

k=0

Now replacing y by p l (= 8"1), we see that

(4)

. Wf)

an+pt(f) = - X

j=l V J J

(3.1-2)

=

P^YP 1 !

-X

j=i v J y

+

XC"1)

(k) f ( k + P l )" an(f)

k=0

V^

an+j(f) + X(-!) r k=o

Since | f(k+p f )-f(k) I p < p" s for all k, |

P

| p < p" 1 for j = 1

and I an(f) I p < 1, the strong triangle inequality shows that (5)

I a n+P t(f) I P < max(p - 1 1 a n +i(f) I p,...,?" 1 1 an+pt(s)_i(f) I p ,p" s ).

p l -1

3.2. Interpolation series in p-adic fields

77

In particular, | an(f) I p < p"1 for n > p 1 ^ . Using this inequality for n > p 1 ^ , we have | an+pt(2)(f) | p < p~2 if n > p t ( 1 ) . In other words, | an(f) | p < p"2 if n > p ^ + p 1 ^ . Repeating this process m times, we have | a n ( f ) | p < p " m if n>p t ( 1 ) +p t ( 2 ) 4---+p t ( m ) This shows the desired assertion: | lim an(f) | p = lim | an(f) I n = 0. We have n—>oo

?

n _>oo

^

proven the following theorem due to Mahler [Ma] for A = Op: Theorem 1. Let A be a closed subring of F r Then we have (i) For each function f : Z p —> A, an(f) e A; (ii) A function f: Z p —> A is continuous if and only if lim an(f) = 0. In this case, the interpolation series E°°_ a n(f)( n ) converges to f(x) for all x e Z p . Proof. We have already proved the assertion (i) and the sufficiency in the assertion (ii) for A = Op. The necessity for (ii) is obvious because the interpolation series converges uniformly on Z p and coincides with f on the dense subset N in Z p . The sufficiency for arbitrary A can be shown as follows. Since Z p is compact and f is continuous, f is bounded. Thus we can find a sufficiently large integer a > 0 such that p a f has values in Op on Z p . Then we apply the result already proven for A = Op to p a f and recover the result for general A. We write O(Zp; A) for the space of continuous functions on Z p having values in A. Since Z p is compact, for f e C(Zp;A), f(Zp) is a compact subset of A [Bour2, 1.9.4] and hence is bounded. Thus the uniform norm | f | p = Sup x6 z p I f(x) I p is a well defined real number. This norm satisfies the strong triangle inequality. Since the uniform limit of continuous functions is again continuous, the A-module C(Zp;A) is complete under this norm (i.e. it is a Banach A-module). Corollary 1. We have | f | p = Sup n | an(f) \v for f e C(Zp;A). Proof. By definition of an(f), we have

I an(f) | p = I ^ L o H ^ G ^ a O ^ Iflp. On the other hand, for each x e Z p m

m

lf(*)lp= l i Hm o Xa n (f)(J)| p = i Bm o IXa n (f)(ylp < Supn | an(f) I p. n=0

n=0

This shows that I f I p < Supn | an(f) | p and the result.

78

3: p-adic Hecke L-functions

§3.3. p-adic measures on Zp In this section, we introduce the theory of p-adic measures supported on Z p . In later sections, we generalize this notion to measures on arbitrary profinite groups. The theory of p-adic measures was initiated by B. Mazur in the 70's in order to construct p-adic standard L-functions for the algebraic group GL(2) ([Mzl] and [MzS]). Increasingly the usefulness of the theory is recognized by every mathematician working with p-adic L-functions because of its conciseness and its relevance to the Iwasawa theory. Let A be a closed subring of F^. An A-linear map (p : O(Zp; A) —> A is called a bounded p-adic measure if there exists a constant B > 0 such that I q>(<|>) | p < B | <|) | p for all <|> e C(Zp;A). Let Meas(Zv\A) be the space of all bounded p-adic measures on Z p having values in A. Then we can define the uniform norm on Meas(Zp;A) by

I 9 I P = S u p | ^ | p = 1 1 (p((|)) | p = S u p ^ 0 ( | cp((|)) | p /| (|) | p ) . Obviously, Meas(Zp;A) is a p-adic Banach A-module under this norm. In particular, if tym converges uniformly to 0 as m —> °°, we see that

I q>(4>m)-) I p ^ I

nT I p and thus lim (p(m) = 9()- W e sometimes write (p(<|)) as J(t>d(p = Jzp<|>d(p. Since each continuous function <>| has a unique expansion into an interpolation series (|)(x) = X ° ° - o a n ^ \ n )

ky Mahler's theorem (Theorem 2.1)

with

an(<|>) e A satisfying H m ^ a ^ ) = 0 and since (J)m = £™=oan(<|>)(n) converges to $ uniformly, we know that

Thus the measure 9 is determined by the sequence of numbers { J ( * ) d c p | n e N} c A . This sequence of numbers is bounded because I p.

Conversely if a sequence of bounded p-adic numbers b n in A is given, the infinite sum Eoo_0bnan(cl)) converges absolutely because of the strong triangle inequality (1.3.1) and lim an((|)) = 0. Thus we can define a measure 9 by

3.3.

p-adic measures on Z p

79

Note that

i j^dcp i p = i xr = 0 b n a n ( ( i ) ) i p < max n I bnan( I p by Corollary 2.1. Thus cp is a bounded measure having values in A and | (p | p < Sup n | b n I p . Since (nj

= 1,

|f

n

) | p = 1, and thus we conclude from J ( n )d(p = b n

that | (p | p > I b n I P for all n. This shows that I (p I p = Supn I b n | p . We have proven the following fact noted by Katz: Theorem 1. Given a bounded sequence {bn} of numbers in A, we can define uniquely a bounded p-adic measure (p satisfying J ( n Jd(p = b n for all n by

All bounded measures on Z p are obtained in this way.

Moreover we have

I 9 I p = Sup n I b n I p. Since (n j is a polynomial of x with coefficients in Q, the value J ( n Jd(p is uniquely determined by the values of Jxmd(p for all m > 0. Thus we record Corollary 1. Each bounded p-adic measure having values in A is uniquely determined by its values at all monomials x m (m = 0,1,2,- ••)• We should remark that the measure corresponding to an arbitrarily assigned bounded sequence of b n is not necessarily bounded: for example, if we assign the value of a measure 9 by f mmj f 1 if m = 1,

Jx dcp = (

then we see easily that

0 otherwise>

c(x\ n J

(-l)P d(p = ——— and thus, (p is no longer bounded. vP ) p

Exercise 1. Give a detailed proof of the above fact. If we assign the value ](n)d(p

am to Jx m d(p

for a e Z p , then naturally

= ( n J e A, whose absolute value is bounded by 1. Thus in this case,

the desired bounded measure exists and is called the Dirac measure at a (i.e. the evaluation of functions at a).

80

3: p-adic Hecke L-functions

Exercise 2. Suppose that Jf(x+y)d(p(x) = Jf(x)d(p(x) for all y G Z p and f G C(Zp;A). Show that (p = 0 if 9 is a bounded measure.

§3.4. The p-adic measure of the Riemann zeta function Let £(s) (s G C) be the Riemann zeta function defined in §2.1. We already know that £(-m) e Q for m e N. In this section, for each positive integer a> 2 prime to p, we show the existence of a p-adic bounded measure £a on Z p having values in Z p such that JxmdCa = (l-a m+1 )C(-m) for all m G N. If such a measure exists, it is unique by Corollary 3.1. As in §2.1, for the function £, : Z --» Z given by

W

1 if n ^ O mod a, if n = 0 mod a,

= 1[l-a

we consider the function (la) Then by Theorem 2.1.1, we have (l-a m+1 )C(-

(lb)

In order to show the existence of the measure £a> writing ( nj = Z£ =o c n! with cn,k G Q, we need to prove that, for all n G N,

I J Q d C a I P = I JZ=o cn,m(l-am+1)C(-m) I p = I 3 n Y(t) 11=1 I p < 1, where we write, as a differential operator,

To show this boundedness, we prepare with two lemmas:

Lemma 1.

tndn We have a n = ~T—rn! dln

3.4. The p-adic measure of the Riemann zeta functions

81

Proof. The assertion is clear for n = 0 and 1. We prove the assertion in general by induction on n. Thus, assuming the above formula is true for n, we compute 3 n+ i. An easy computation using the definition of the binomial polynomial shows that / x \ (x-n) . / x \ n!

U l j = oST)! W-

Then we see that 1

| which proves the assertion.

By Leibniz's formula, we see that dn(fg) = y n / n \ drf d n " r g Z , r = 0 W d t r dt n-r • dt n Multiplying both sides of the above formula by tn/n!, we get (2) Lemma 2. Let

3 n (fg) = X^ = o Orf)On-rg). Rf = { - ^

| P(t), Q(t) e Z p [t] and | Q ( l ) | p = l } . Then

R1 is a ring and is stable under 3 n for all n. P P' Proof. Taking Q and QT from R', we have PP' P P ' PQ'±P'Q P P1 7 G R and x 76 R Q Q " QQ' Q Q7 = 1 which shows that R is a ring. We now show R' 3 9nR' by induction on n. When n = 1, we see that

which shows the result. Now assuming the result is true for dm for m < n-1, we shall show R' 3 dnR\ Applying (2) for f = Q and g = Q - 1 with

P/Q G R\ we have X"=o^ r ®^ n - r ( 5 ^

=

°*

Since 9

° ^= ^'

we have

By the induction hypothesis, 3n-r(Q"1) e R' for r > 1. Since R' is a ring containing Z p [t] s 3 r Q, the above formula shows that 3n(Q"1) ^ R'- Then again by (2), we have

82

3: p-adic Hecke L-functions

Since 9n(Q"1) e R', again by induction assumption, we see that dn(-Q) e

R\

We are now ready to prove Theorem 1. Let a e N with a > 2 and (a,p) = 1. Then there exists a unique bounded p-adic measure £a on %p having values in Z p such that

JxmdCa=(l-am+1)C(-rn) far all me N. Proof. As already remarked, we only need to prove

U(n>Calp= k By definition, we see that

and hence *F e R'. Thus by Lemma 2, we see that d^ P

i

e R'. This implies

i

3 n ¥ = Q for P, Q e Z p with | Q(l) I p = 1. Thus we see that P(l) e Z p and

§3.5. p-adic Dirichlet L-functions Since the absolute value | | p is an ultra metric (i.e. a metric satisfying the strong triangle inequality (1.3.1)), Z p is totally disconnected (i.e. Z p is a disjoint union of arbitrarily small open sets). In fact, if we write D(x,p" r ) = { y e Z p | | y - x | p < p - r } = {y€ Z p | I y-x I p < p" r + 1 }, this is an open set and Z p = U p D(x,p~r) (disjoint union), because D(x,p- r ) = { y e Z p | y = x mod p r Z p } . By this, there exist locally constant but non-constant continuous functions. We compute in this section the integral of such functions with respect to d£aIt is clear from the argument in §3.4 that to each F(t) e R1, we can associate a p-adic measure |ip on Z p with values in Z p in the same way:

3.5. p-adic Dirichlet L-functions (la)

/ © ^ F

=

83

( 9 nF)(l) = 5 f ^ ( D for all n e N.

Now expanding F into its Taylor expansion around t = 1, we see that )T" (for T = t-1). In this way, we can embed R' into Zp[[T]]. Conversely if a p-adic measure 9 on Z p having values in Z p is given, then we can define the corresponding power series <E>cp(t) by

<M0 = £1=0 (J(n)d. Thus the map 9 H» O(p induces an isomorphism: Meas{Zp\Zv) = Z p [[T]]. Choosing a basis {wi,...,w r } of Op over Z p , for each measure 9 in Meas(Zp,Op) and for <|> e (XZv;Op), we can write 9() = 9i((|>)wi+---+9r((|))wr. Then it is plain that q>i e Meas(Zp;Zp). Thus we know that Meas(Zp;Op) = p Thus we know that (lb) the map cp h-> Ocp induces an isomorphism: Meas{Zp\O^) = Op[[T]]. Exercise 1. Show Jxmdcp = (t^)mOcp | T =o (T = t-1) for all m e N. When I y I p < 1 ( y e Op), the following infinite sum is convergent: (1+y)X=ir=o G VforxeZp. Thus we define the p-adic power yx (x e Z p ), when | y-11 p < 1, by

yx = X n = 0 W ( y ' 1 ) n

(2a)

for XG Z

P*

This definition coincides with the definition given in §1.3 because the two definitions coincide with each other if x is a positive integer. It is easy to check that the usual exponent rules hold: y x+z = y x y z , y x z x = (yz)x and y° = 1 because of the density of N in Z p . Using this trick, we know that oo

f

.

f/'xN

i

i

(2b) J y x d 9 ( x ) = £ J ( J 1 J d 9 ( y - l ) n = *q>(y) for y e Op if I y-1 \ p < 1. n=0

Exercise 2. For two measures 9 and \\f in Meas(Zv;Op), define the convolution product 9*\|/e Meas(Zv;Op) of 9 and \|/ by

84

3: p-adic Hecke L-functions

Jfd(q>*\|0 = JJf(x+y)d(p(x)d\|/(y). Show O (p *y = O ( p O v in O p [[T]]. The space of measures 5Vfe&?(Zp;0p) is naturally a module over the ring O(Zp;0p) in the following way: for f e C(Zp;Op) and 9 e

Meas(Zp;Op),

We want to determine Ofq> for some special f supposing the knowledge of <Xy First let us deal with the function f(x) = z x for z e Op with | z-1 | p < 1. By (2b), we see that ttfip(y) = Jy x dfcp(x) = Jy x z x dcp(x) = J(yz) x d(p(x) = *
Op with

\z-l\

p<

I, we have O zX#q) (t) = ©
Op[[t-l]].

Let jipii be the group of pn-th roots of unity in an algebraic closure of Fp. Let K = Fpfjipn] be the field extension of F^, obtained by adding all elements in jipn. Let R be the integral closure of Op in K. Then, as seen in §1.3, there is a unique norm | | p on K extending that on F. Since there are no p power roots of unity except 1 in characteristic p, £ mod & = 1 for the maximal ideal (P. Thus I £-1 I p < 1 for any £ e |j,pn and hence we can think of £ x for x e Z p . If x E k m o d p 1 1 for k e N, we have £x = £ k because Cp" = 1. If § : Z p /p n Z p = Z/p n Z -> K, then

4>(x) = p " n I ^ ^J?

(4)

This follows from the orthogonality relation V rx"b = J p n Zrf?€p.pn^ [0

if x = b mod p n Z p , otherwise.

Thus, applying (3) for z = £ and writing b

we see that, for a locally constant function Z/pnZ

(i.e.

(() = (()'op

for $ e CCZpjO^,) factoring through

for the projection

p : Z p —» Z / p n Z

n

(|)' : Z / p Z -> O p ), (5)

<E>4>q> = M>]9 for (pe f^eo5(Zp;Op)

and

with

3.5. p-adicDirichletL-functions

85

Here note that [^JO^ actually belongs to Op[[t-1]] because of (5), although this fact is not a priori clear from the definition of [(t)]O
(6a) (6b)

^

|

if O(£t) = O(t) for all £ e ^ipn, then [<|>](O0) = ([<|>]0)O,

where O, 0 e Op[[t-1]]. In fact,

where we have again used the orthogonality relation. Since tetm = mt m , the second formula of (6a) is obvious from the first. As for (6b), we compute

= *(t)[«e(t).

Exercise 3 . Let R = { ] | | | P(t), Q(t) e Z p [t] and | Q ( l ) | p = l } . Then show that R 3 [<|)]R for a locally constant function <|>: Z p —> Z p . We know from the definition (4.1a) that

Using the geometric series, we see that —- = l+t a+t2a+---+tna+---. Therefore apn m=l

b=l

m=0

l-tap

We put 0(t) = ££fi£(b)t b and O = l/(l-t apn ). Then we have W =
By (6b)

m=l

m=l

m=l

Writing (J)a(m) = <|)(am) and supposing (|) actually has values in FnO^ (and hence has values in C 3 F), we can consider the complex L-function

86

3: p-adic Hecke L-functions oo

L(s,$-a k+1 (t) a ) = X (
As seen in §§2.1-3, this sum is convergent in C if Re(s) > 1 and has a meromorphic continuation to the whole complex plane, and

If % is a character of (Z/p n Z) x having values in F x , extending % by 0 on Zp-Zpx and denoting this function by the same symbol %, we have a locally constant function % o n Z p . Then L(-m,x-am+1%a) = (l-am+1%(a))(l-%0(p)pm)L(-m,x0), where %0 = % if % is non-trivial and %0 is the constant function on Z p having value 1 if % is trivial. Thus we have proven the first part of Theorem 1. If % : (Z/pnZ)x —> F x is a primitive character {when % is trivial, we agree to put n = 0), then J%(x)xmdCa(x) = (l-a m+1 x(a))(l-x 0 (p)p m )L(-m,x 0 ) for all m e N. If % is an even character x (i-e- %(-!) = 1)> f ^ if x = i d , a W x ) \.(l-Z(a))p- n G(z)X bG(Z/pnZ)X x- 1 (b)lo g (l-C b ) if X * id, where G(%) = E ^ x C b ) ^ (the formula is independent of the choice of Q. To finish the proof of the above theorem, we need to compute J%(x)x"1d^a(x). By Corollary 2.3.2, if %(-l) = -1, the integral is identically 0 and is not included in the theorem. Thus we assume that % is even. We first assume that | a-11 p < 1.

fi-ta] Writing O(t) = log-j

k we can expand it formally as a power series in

fi-t a l i T = t-1 in Q P [[T]] because < > =a, which is in the domain of 1 [1-tJ tconvergence of the p-adic logarithm. Write this power series as G>. Then inside Qp[[t-1]], we have from (6a) and (7) that [%W = t^[%]O = ( l + T ) ^ * ] * . On the other hand, since % is supported on Z p x , the function x h-> x ' ^ x ) is a continuous function on Z p . Thus we may consider the measure (p given by xp = J(|)(x)x-1x(x)dCa(x) for all 0 e C(Zv;Op).

3.5. p-acfo DirichletL-functions

87

By the definition of O

Jx"1%(x)dCa(x) = Now suppose that % | i+pn-izp * 1 (i.e. % is primitive modulo p n ). Then by Lemma 2.3.2, we see, writing £ = e(—), that P Therefore [%](Cc) for C = e(-±j). When is the trivial character, we see that 1 fMrbIP" =

V

l-l

Zb

if

?

=

if £ * 1.

Using the extended log in (1.3.9b), we can obtain the formula in the theorem when I a-1 I p < 1. The formula holds for general a because the p-adic L-function defined below is independent of a. Let us define p-adic DirichletL-functions. Since %(x) is only supported on Z p x , we may write JZpXX(x)xradCa(x) = (l-a m+1 x(a))(l-x 0 (p)p m )L(-m,x 0 ) for all m € N. We can decompose Z p x = |a. x W where W = l + p Z p for p = F

p

r4 if p = 2, . u IP otherwise,

and |i is the maximal torsion subgroup of Z p x . Fixing an element u G W-(l+ppZ p ), we have an isomorphism of topological groups Z p = W given by Z p 9 s h) u s e W. Let co : Z p x -> |4, be the projection map. Thus co(x) = Mm x pn if p > 2 and co(x) = ±1 according as x = ±1 mod 4Z2 if p = 2. This character co is called tha Teichmiiller character. Then we define (x) for x e Z p x by (x) = C G M ^ X . Then X H (X) is the projection of Z p x onto W. We then fix a character % of (Z/prZ)x and define the p-adic Dirichlet L-function with character % by

88

3: p-adic Hecke L-functions

Then Lp(s,%) is a continuous function on Z p except when % = id, and in this special case, £p(s) = Lp(s,id) is a continuous function defined on Z p -{1}. Moreover we have the following evaluation formula: (8)

Lp(-mjc) = (l-(%co-m-1)()(p)pm)L(-m,(xco-m-1)0) for all m e N,

where the right-hand side is the value of the complex L-function while the left-hand side is the value of the p-adic L-function and the values are equal in the field F common to ¥p and C. Since the right-hand side is independent of the choice of a, we conclude from the density of N in Z p that Lp(s,%) is independent of a. We now show that Lp(s,%) is a p-adic analytic function on Z p when % * id and a p-adic meromorphic function on Zp-{1} when % = id. We can define a p-adic measure £a,% o n W (= Z p ) in the following way. For any given continuous function <>| : W —» Op, we define

J w (Ky)dU x (Y) = JZpX%co-1(x)0((x))dCa(x). Then we can write Lp(s,%) = (l-%(aXa)1~s)"1JwY~sdCa,%(Y)- Identifying W with Z p via i : Z p s s (-) u s e W, we can associate with % a power series 3>a,%(t) G Op[[t-1]] as in (3) in the following way:

(l- % (a)(a) 1 - s )L p (s,%) = J w y sdCa,x(Y) = JZpu"sxd(t*Ca,%)(x) = * q > ( O . We define tt^t) by Q9(v'h). Then Lp(l-s,%) = (l-%(a)(a) s )- 1 O aa (u s ). By (1.3.6a,b), the exponential (resp. the logarithm) function converges on pZ p (resp. W). Thus we can write us = exp(slog(u)). This shows that Lp(s,%) is a meromorphic function on Z p whose pole comes from the possible zero of (1-X(a)(a)1"s) at s = 1. When %(a) * 1, this function (l-xCaXa)1"8) does not have a zero at s = 1 and hence Lp(s,%) is an analytic function. Here note that Lp(s,%) itself does not depend on the choice of a because of the density of N in Z p and the evaluation formula (8). Since Z p x = |i.xW and W is topologically generated by u and jn is cyclic (When p = 2, |i = {±1}, and when p > 2, [I is the group of(p-l)-th roots of unity), we can take a so that co(a) generates L |L and (a) topologically generates W. Then %(a) = 1 <=> % = id. Thus if % *id, Lp(s,%) is analytic on Z p . We see that (a) s = l+log((a))s+higher terms of s by (1.3.8a,b). This shows that £p(s) has a simple pole at s = 1 whose residue is (1-p"1). Summing up these considerations, we have

3.6. Group schemes and formal group schemes

89

Theorem 2. For each Dirichlet character % : (Z/p r Z) x -* F x with %(-l) = 1, there exists a p-adic analytic function Lp(s,%) on Zp-{ 1} such that Lp(-m,x) = (l-(xco-m-1)0(p)pm)L(-m,(%co-m-1)0) for all m e N. % w non-trivial, Lp(s,%) zs analytic even at s = 1 <9« r/ze other hand, £p(s) =Lp(s,id) to a simple pole at s = 1 vv/jose residue is d-p- 1 ). When % is an odd character (i.e. %(-l) = -1), then xco111"1^!) = (-l) m because of co(-l) = -l, and thus L p (-m,x) = 0 for all m e N by Exercise 2.3.3. This shows that Lp(s,%) is identically zero if % is odd. This is why we excluded the case of odd characters %. Although non-trivial Lp(s,%) exists only for even %, the values of complex L-functions with odd characters show up as the special values of the p-adic L-function Lp(s,%). In fact, when m is even, %CGml is an odd character. §3.6. Group schemes and formal group schemes In order to give an interpretation of p-adic measure theory on Z p using formal multiplicative groups in the next section, we recall here briefly the properties of affine schemes and affine group schemes. For details of the theory, see Mumford's book [Mml, §11]. Let A be a commutative algebra with identity and R be an A-algebra. We consider the affine scheme G/A = Spec(R)/A- Thus G is a covariant functor from the category J%C#/A of A-algebras to the category of sets given by G(S) = HomA-aig(R,S) for any A-algebra S. For any algebra homomorphism (p : S —> T, the functorial map G((p) : G(S) —> G(T) is given by G(9)((()) = (po(|). The set G(S) is called the set of S-valued points (or simply S-points) of G. For two affine schemes GA and G'/A = Spec(R')/A, a morphism of schemes f : G —> G' is a set of maps f(S) : G(S) -> G'(S) such that for every A-algebra homomorphism 9 : S -» T, the following diagram is commutative: G'(S)

Let Scfi/A be the category of schemes over A [Ha, II]. If r\ : R' -> R is an A-algebra homomorphism, then T|* : G —> G' given by ri*(S)((()) = §°r\ is obviously a morphism of affine schemes. Conversely if f: G —> G1 is a morphism of affine schemes, we in particular have a map f(R): G(R) -> G'(R). We have two natural maps:

90

3: p-adic Hecke L-functions i : HoiriA-aig(R\R) -> H o m ^ C G ' ) given by r| h-> r|* and K : Hom^CCG') -» HoiriA-aig^R) given by jc(f) = f(R)(id).

By definition Ti*(R)(id) = id<>r| =r|. Thus 7uoi = id. Let T] = f(R)(id) for the identity map i d e G(R). We want to show f = r|* (i.e. 107c = id). We have a commutative diagram (for any map <|>: R -> S of A-algebras): G(S) t Gfl>)

f(S)

) G'(S) T G'(4>)

This implies

This shows that (1)

71 : Hom^(G,G') s HomA-aig(Rf,R) by n(i) = f(R)(id).

Let G = Spec(R), G1 = Spec(R') and S = Spec(B) be (affine) A-schemes and f : G —> S and g : Gf —> S be two morphisms of A-schemes. We consider the following universal property for an A-scheme X. Whenever we have a commutative diagram T —*-* G G-

- ^ s ,

we have a unique morphism of schemes h : X -» T such that fo(()oh = gocpoh and p = <|)°h and p' = cpog are independent of 9 and <|). This universality is the contravariant version of the universal property defining the tensor product R<E>BR' (see §1.1). Thus such a universal object X is given by Spec(R®BR'). We write X = Gx s G'. Now we suppose that R has the structure of an A-bialgebra; that is, there are three A-algebra homomorphisms m : R -» R® A R, e : R -> A and i : R -> R satisfying (Gl) The diagram

R —^—» R® A R R(2>AR

m®id

is commutative;

3.6. Group schemes and formal group schemes m

(G2) The diagrams

91

m

R —> R® A R and R —> R®AR are commutative; id \ j

^ id®e

id \

R

R

m

(G3) The diagrams

^ e®id

m

R -> R® A R and R -> R®AR are commutative, A -> R®A R

A-> R®AR

i R = R R = R where JLL : R®A R -> R is the multiplication: a®b h-> ab; m

(G4) The diagram

R -> R® A R 3 x ® y m^i

is commutative.

X

1

R® A R 3 y®x Since GxAG(S) = HomA_aig(R®AR,S) = HomA.aig(R,S)xHomA.aig(R,S) = G(S)xG(S), the morphism m (resp. i) induces m* : G(S)xG(S) -» G(S) (resp. i* : G(S) -> G(S)), and e : R -> A induces e e G(S) by composing the original e with the A-algebra structure: A —» S. Writing x.y e G(S) for m*(x,y) (x,y e G(S)) and x"1 for i*(x), we know that (i) (x.y)-z = x.(y.z) (Gl), (ii) x.e = e.x = x (G2), (iii) x -1 .x = x.x"1 = e (G3) and (iv) x.y = y.x (G4). This shows that G(S) is an abelian group. In particular, we have e = e = i*e = e°i. Thus if R is an A-bialgebra, then G is a functor on Sch/A having values in the category &6 of abelian groups. In this case, G is called a commutative group scheme defined over A. When R only satisfies Gl-3 but not G4, G(S) is a group (which may not be commutative). In this case, we just call G a group scheme. A morphism of group schemes G and G' is a morphism of schemes which induces a group homomorphism between G(S) and G'(S) for all S. It is easy to check under the isomorphism (1) that such a morphism corresponds to a homomorphism cp of A-bialgebras, i.e., cp : R' -» R is a homomorphism of A-algebras which makes the following diagrams commutative: for the morphisms of bialgebras mf, e' and i1 (resp. m, e and i) for R' (resp. R) m

R

i R

i

9

.

• R'®ARf

icp®9 R® A R,

R

i R

1

9

e ->

R1 U R1

A

1 A

and

I (p i cp R -> R.

92

3: p-adic Hecke L-functions

Conversely if a functor T : Sch/A -> A& is given and if T(S) = HomA-aig(R,S) for every A-algebra S, we can define an A-bialgebra structure on R by (1) using the group laws (i)-(iv) above. Thus any functor from SCUJA to Sib represented by an A-bialgebra R is in fact a commutative group scheme. Let cp* : G —> G1 be the homomorphism of group schemes induced by (p : R' —» R. We then consider R

). We now have an exact sequence of groups for all S: 1 -> Ker(q>*)(S) -> G(S) Exercise 1. (i) Give a detailed proof of the fact that R 9 is an A-bialgebra. (ii) Prove the above sequence is exact. Let us now give some of the most important examples of commutative group schemes. Let A = Z and R = Z[t,t"1] for an indeterminate t. We define m : Zfot" 1 ] -» Z[x,y,x" 1 ,y" 1 ] = R® A R by m(t) = xy, i : R -» R by i(t) = f1 and e : R —» Z by e(t) = 1. Then it is easy to check properties Gl-4 and thus R gives a group scheme defined over Z. We write this scheme as G m (or GLi). Any algebra homomorphism of R to a ring S is determined by its value at t. Thus G m (S) can be embedded into S. If x e G m (S), then x(t" 1 ) = x(t)" 1 G S and therefore G m (S) = S x . If x,y e G m (S), then x»y(t) = x(t)y(t) by definition, and thus Gm(S) is the multiplicative group S x as a group. We consider the endomorphism [N]s of G m (S) given by [N]s(s) = sN. This is induced by an endomorphism [N] of R taking t to t^. Then |i N = Ker([N]) = Spec(Z[t]/(tN-l)) and hence

M S ) = {?6 s x U N = i ) . Let G/A = Spec(R)/A be a commutative group scheme. We may regard 0 G R(8>AS as a function on G(S) having values in S for any A-algebra S in the following way: <|>(s) = s((|)) for s G G(S) = HomA-aig(R,S) = HomS-aig(R®AS,S). We then consider the space DerA(G) of derivations on R over A having values in R. Thus DerA(G) is an R-module of A-linear maps D : R -> R satisfying D(fg) = fD(g)+gD(f). The A-linearity of D is equivalent to the fact that D annihilates A in R. The composition of two derivations is no longer a derivation, but we still can think of the algebra Diff(G) generated by derivations over A in An element of Diff(G) is called a differential operator on G. A dif-

3.6. Group schemes and formal group schemes

93

ferential operator D : R —> R is called invariant if the following diagram commutes: R-^R (2a) im im

R® A R -> R<8>AR. D®id

We write £(G) for the A-algebra of invariant differential operators. A differential operator D is invariant if and only if for any <>| e R, defining (|)x = (id®x)om(<|>) e R® A S for x e G(S) = HomA_aig(R,S), we have (2b)

D(<|)x) = (D(|))x for all x e G(S) and all S /A .

In fact, for y e G(S), §x(y) = (y®id)o(id®x)om(<|>) = (y<8)x)om((|)) = <|)(y»x). Thus (|)x as a function on G(S) is the right translation of <> | . Thus it is legitimate to call D invariant if (2b) holds for all S and all x. Let us prove the equivalence of the conditions (2a) and (2b). The commutativity of (2a) implies D((|)x) = (D®id)((id®x)om(<|>)) = (id®x)o(D®id)om(<|>) = (id®x)om(D<|>) = (D<|>)x. Conversely if the above identity (2b) is true for all x, then it is obvious from (1) that the diagram (2a) is commutative and D is invariant. We write £teA(G) for the space of invariant derivations on G/A. If D is a derivation on Gm/z over Z, it is determined by its value at t (because R = Z[t,t"1]). On the other hand, D' = D(t>3r is a derivation satisfying D'(t) =D(t). This shows ) = R ^ and Derz(Gm/z)zC = Der c (G m / c ), and thus Liez(Gm) injects into £tec(Gm). Since a Gm-invariant first order differential operator on G m (C) = C x in Cfot^br is, by (2b), a constant multiple of tjjj: = dZ^(t)» w e (3a)

k n o w that

Lie

z(Gm)

= z^f-

Thus

Der z (G m ) = R-|, LieZ(Gm) = Z t | and #(Gm)zQ =

^

A polynomial on Z is called numerical if P(n) e Z for all n e N . We know from Proposition 1.1 that the binomial polynomial gives a basis over Z of the ring of numerical polynomials. Thus we have by (3a) that (3b)

£>(Gm) = {P(tgjr) I P(T) is a numerical polynomial} = Z[3 n ] nG z,

94

3: p-adic Hecke L-functions

where 3 n is the differential operator in Lemma 4.1. Writing Dp = Pter), we see easily that Dp is characterized by the fact Dp(tm) = P(m)tm. We consider in this note a group scheme a little more general than Gm. Let M be a free Z-module of finite rank. We consider the group algebra R = Z[M]. Thus R is a commutative ring generated by a Z-free basis t a for a e M (t° = 1) and t a t p = t a + p . When M = Z as an additive group, R = Zfot' 1 ]. Note that any algebra homomorphism X : Z[M] —> S is determined by its value at t ai for a basis {cci} of M. Since t ai is invertible in Z[M], ^(t ai ) has to have values in S x . Thus, writing T for Spec(Z[M]), we know that (3c)

T(S) = HomA-aig(Z[M],S) = Hom gr (M,S x ) = Hom gr (M,G m (S)).

Thus as a functor, we know that T = Hom gr (M,G m ). Since T is a group functor, by (1), T is an affine group scheme. Exercise 2. Make explicit the bialgebra structure of Z[M]. If we identify M with Z r choosing a basis

{OCJ},

we have an isomorphism

This shows that T = G m r non-canonically. We call a rational polynomial P on M numerical if P(a) e Z for all a e M. Then by (3b), we see that (3d)

2XT) = {Dp IP is a numerical polynomial on M),

where Dp is the invariant differential operator on T given by D P (t a ) = P(a)t a for all a e M. Let K be a finite extension of Qp, and assume A to be the p-adic integer ring of K. Hereafter, we always consider T to be defined over A. Thus we change the notation and write hereafter R for the coordinate ring A[M] of T over A. For each integer N, we define the endomorphism [N] which takes x e T(S) to x N G T(S). Then as a group subscheme of T, we define T N = Ker[N] in T. Then T N = Spec(RN) for RN = Z[M/NM] = ®iA[t a \f ai ]/(t N(Xi -l), and we see that (4) TN(S) = HomA_aig(A[M/NM],S) = Homz(M/NM,|iN(S)) for all S. We now restrict the category on which the group functor T is defined. Let AcC = JZCC/A be the category of A-algebras S = lim (S/paS) for the maximal ideal p of A. Let 9&C/A be the subcategory of &<£ consisting of A-algebras in

3.6. Group schemes and formal group schemes

95

which p is nilpotent. Thus we may regard the category Ad of /?-adic algebras as the category obtained from 9{iC adding projective limits of its objects. Let F = Spec(R')/A for an A-algebra R'. We consider the restriction of F to fAfif. Then for any S e Ob(fA#0> every x e F(S) = HoiriA-aig(R\S) has to factor through R7paRf for sufficiently large a (such that pa kills S). Thus F(S) = HomA-aig(R!,S) = Hom A -ai g (R\S), where R' = lim (R'/j^R') = lim (R7paR!) is the p-adic completion of R!. Thus considering the restriction of F to 9& is studying the p-adic completion of R' algebraically and is studying the germs (with infinitesimals) along the special fiber at p of Spec(R'). For S e ObfcAfiO, we write Sred for the reduced part of S, i.e., the residue ring of S modulo its nilradical n$. Returning to the original R = A[M], we define a new functor T : 9{iC -» AS by (5)

f (S) = Kernel of the natural map T(S) -» T(S red ).

Since T(S) = Homgr(M,Sx) for any S, f (S) = Hom gr (M,l+n S ). Since p is nilpotent in S, for any x e ns, taking N so that x = 0 , we see that if p n > N .

Since

P

1

-> 0 as n -> oo if

I i I < N, we see that 1+ns is contained in |apn(S) for large n. On the other hand, if £ e |i p n(S),

rn is divisible by p, because l+i-l)?11 and

for 1 < i < p n are both divisible

by p. Thus (£-1) is nilpotent in S. Thus (6)

If p is nilpotent in S, |ip~(S) = lim M-pa(S) = a This shows that, because p is nilpotent in S, f (S) = Homgr(M,^poo(S)) = Hom conti (M p ,S x ), where M p = lim (M/p a ) and S x is supposed to have the discrete topology, a while M p is equipped with the p-adic topology. Then the last equality follows from (4) because any continuous homomorphism of M p to S x factors through M/p a M for some a. Now we can extend this functor to M by (7a)

f ( S ) = Hm f(S/paS) = Hom conti (M p ,S x ), a

96

3: p-adic Hecke L-functions

where f (S/p a S) = Homconti(Mp,(S/jpaS)x). Thus, by (4), on Ad, we see that f (S) = HomA-aig(RP-,S) for (7b)

Rpoo = lim R p n = lim <8>iA[ta\fai]/(tpn(Xi-l) = A[[t a i -l,...,t a r -l]]. n

n

In this sense, we may write T = Spf(Rp~)> which is the formal completion of T along the ideal (t t t l -l, ..., t ar -l) (see [Ha, II.9]). We thus call this functor the formal group (scheme) of T. To complete the proof of (7b), we need to show lim A[t,t"1]/(tpn-l) = A[[t-1]] as compact algebras. n

Since A f c f 1 ] / ^ - ! ) = A[t]/(t pn -l) = A[T]/((T+l) pn -l) for T = t-1, we first show that (T+1)^-1 e (p,T) n+1 . When n = 0, this is obvious. We proceed by induction on n. We see that (T+l) pn+1 -l = ((T+l) pn -l)(l+(T+l)+---+(T+l) p - 1 ) = ((T+l) pn -l)(p+TQ(T)) for an integral polynomial Q of T, and hence by the induction hypothesis, (T+l) p n + -1 E (p,T) n+2 because (p+TQ(T)) e (p,T). Thus there is a natural map

lim A f c r 1 ] / ^ " - ! ) -> Hm A[T]/(p,T) n+1 = A[[T]]. This map is conn

n

tinuous under the projective limit of the natural topology on both sides. By compactness, the image of the map is closed and contains a dense subset A[T]. Thus the map is surjective. The injectivity of the map is obvious because n n (p,T) n = {0}. This shows the isomorphism. §3.7. Toroidal formal groups and p-adic measures We study here the relation between the space of p-adic measures on M p and the coordinate ring Rp~> of the formal group T. We will get a natural isomorphism Rp«> = Meas(Mp;A) generalizing the isomorphism A[[t-1]] = Meas(7*v\K) given in (5.1b). Here we keep the notaion of the previous section. In particular, the base algebra A is the p-adic integer ring (with the maximal ideal p) of a finite extension K/Qp. We continue to use the notation introduced in the previous section. For S e Ob(.#d/A), we can write Meas(Mp;S) = {cp : C(Mp;S) —> S | cp is S-linear and continuous}, where we use the topology of uniform convergence in the space £(MP;S) of continuous functions on M p with values in S, while S is equipped with the p-adic topology. By definition, we have

3.7. Toroidal formal groups and p-adic measures

97

Rpoo = Mm R p n = Urn A[M/pnM] = A[[t a i -l,...,t a r -l]] n

n

and RPoo<§>AS = Urn (R p oo® A S/p a S) = Mm S[M/p n M] = S [ [ t a i - l , . . . , t a r - l ] ] . a n

For any finite group G, the group algebra A[G] has the obvious universal property: for any given group homomorphism 2; : G -> S x for S e J%Cg/A, there is a unique A-algebra homomorphism £* : A[G] -> S making the following diagram commutative: A[G] —S—»

T G

S

t — ^

Sx

The algebra R poo ® A S has a similar universal property for continuous homomorphisms £ : M p -> S x (i.e. £ e f (S) = Hom con t(M p ,S x )). Thus, for any given £ e T(S), by continuity, ^ a : x H £(x) mod paS factors through M/p aM. Therefore, by the universality characterizing group algebras, we have the S-algebra homomorphism £ a * : S[M/p a M] = Rpa(g>AS -* S/paS extending £ a . By construction, {£a*} forms a projective system yielding £* = lim £ a * : Rpoo<§)AS -» S, which extends £. Thus Rpeo<8>AS is sometimes written as S[[M]] and is called the continuous group algebra of M. Anyway we have recovered the canonical isomorphism proved in the previous section: f (S) = HomA_aig(Rpeo,S) = HomS-aig(RPoo<§>AS,S) (^ h* ?•). In naive geometric terms, the evaluation at an S-rational point % of a variety gives an algebra homomorphism from its coordinate ring into S. Reversing this process, in modern algebraic geometry, we define S-rational points to be algebra homomorphisms from the coordinate ring (given without any reference to geometric objects) into S. From this point of view, the value f(£) of an element f in the coordinate ring ?.s a function of S-rational points £ is nothing but the value of the algebra homomorphism ^ at f. Following this convention in algebraic geometry, we regard an element f = f(t a i -l,...,t a r -l) ai

ar

Rp~AS = S[[t -l,...,t -l]] of

in the coordinate ring

f as a function of £ e f(S) by putting

98

3: p-adic Hecke L-functions

We now want to prove the following result of Katz [K6]: Theorem 1. Let S be an object in Mj^, Then there is afunctorial isomorphism of A-algebras

between

R P OO®AS (= lim (RpooAS//7aS)) and a

f7tfea5(Mp;S)

characterized by the following properties. Writing the correspondence as 3 f <-> L | Lf e Meas(Mv;S), we have

RP<§>AS

(i) for each point £ e f (S) = Hom cont (M p ,S x ),

J^(x)d|i f (x) = f(£);

(ii) for any numerical polynomial P : M -» Z regarded as a continuous function on Mp, where Dp is the differential operator in (6.3b); (Hi) for £ flttd P as above, and for any locally constant function <|>: M p -> S, /or F(t) = ([<|)]f) /or the operator [<|)] on Rpoo^AS satisfying [<\>]ta = (|)(a)ta. Before proving the theorem, we recall briefly the isomorphism (5.1a), its construction and its properties. By Mahler's theorem (Theorem 2.1), to each measure (p e Meas(7jv\S), we assign a power series

(•)

O(t-l) = X J(J)d
where in the last equality, we regard d|J, as a measure having values in S[[t-1]] in oo

an obvious manner, using the expansion tx = ^

(

n

Vt-l) n . Write O = O 9

n=0

and (p = (p^>. Then we have verified (la)

for each £ e

G m (S), J§(x)dq> = * 9 (?(1)-1) = ® 9 (§),

where in the last equality, we have used the convention described above the theorem. Since Z p is topologically generated by 1, we see that ^(x) = x . Thus replacing y in (*) by ^(1), we know that

3.8. p-adic Shintani L-functions of totally real fields

99

We already know from (6.3a) that any invariant differential operator of Gm/z is given by Dp = P(tjr) for a numerical polynomial P. Then we see from Exercise 5.1 and (la) that db) Finally, for any locally constant function <|>: Z p -» S, we have (lc)

J]O
where [(()] is the operator defined in (5.6a,b) satisfying Proof. We may identify T = G m r and M = Z r by fixing a basis {(Xi) of M. Writing ti for t ai , we already know that Rp~ = A[[ti-l,...,t r -l]] (6.7b). Then writing n = (nO for an r-tuple of natural numbers (i.e. n e Nr) and defining (nJ

= n i=i( n !) f° r

x =

(xi) G Z p r

=

M p , we see in the same manner as in

§3.3 that (2) Any f e C(Zpr;S) can be expressed uniquely as an interpolation series: f(x) = S n >oa n (f)( n )

with

\ii/

means

£ n = o * " ^ n = o and

satisfying

a

n(f) e S satisfying

Mm an = 0, where I n > 0 Inl—>©o

| n | = E i l n i l . Conversely,

if a sequence

l i m a n = 0 is given in S, then the infinite sum S n > o a n ( n ) Inl—>°°

{a n } con

"

\ii/

verges in S giving a continuous function on Z p r having values in S. In fact, if we write £O(Mp;S) for the space of locally constant functions on M p with values in S, it is plain that LC(Zvr;S) = LC(Zv;S)®s--®sLC(Zp;S). Since XO(Zp;S) is dense in O(Zp;S) under the topology of uniform convergence, we see that C(Zpr;S) = C(Z p ;S)®s---®s£(Z p ;S). Then (2) follows from Mahler's theorem. Then all the assertions of the theorem are easy consequences of (la,b,c).

§3.8. p-adic Shintani L-functions of totally real fields We fix an algebraic closure Q p of Q p and Q of Q. We also fix embeddings of Q into Q p and C so that any algebraic number can be regarded both as a complex number and as a p-adic number. We extend the absolute value I I p as assured in (2.5). Let F be a number field of degree d in Q. We write I for the set of all embeddings of F into C. We assume that all the embeddings of F into C actually fall in R, i.e., that F is totally real. We write R+ for the strictly positive real line. We write FR = F ® Q R , which is canonically isomorphic to R1 via F 3 \ h-> (t,c)oeh We put FR+ = R+1. For any subset X of FR, we

100

3: p-adic Hecke IAunctions

write X + for Xf!FR+. We write E for O+x = O T | F R + . Let v = {vi, ..., vr} be a set of Q-linearly independent element in F + . We write C(v) for the open simplicial cone spanned by v, i.e., C(v) = ZiR+Vi in FR+. AS shown in §2.7 (Theorem 2.7.1), there exists a finite set V of the v's as above such that FR+ = UEG EUVG V£C(V) (disjoint union).

Let O be the integer ring of F. We fix an ideal a* {0} in O. For any X e F + , we can replace V by XV = {Xv | v e V} without affecting the above property. In particular, we may assume that v is contained in a for all v e V. We consider the torus T/z associated with M = a defined in (6.3c). Let R be the coordinate ring of T. Then R is a group algebra of a, i.e., R = Z [ t a | a G a]. Let Z p be the integral closure of Z p in Q p . We define R ' P = { ^ § | P ( 0 , Q ( t ) e R<8>zZp and Similarly, we define ^ |

R<S>ZC and

We put R(v) = o+ntZvievXiVilo < xi< 1}. For each v e V, x e R(v) and for £ e T(C) = Hom(a,C x ), we define

da) (lb)

When ^(vi) * 1 and | ^(vO I < 1 for all vi e v, we see that

(lc)

If ^(vO is a non-trivial / m-th root of unity for a prime / prime to p (m > 0), then | ^(vO-1 I p = 1 and hence fv^,x e R'p.

For the coordinate ring Rp°o of the formal group f, the argument which proves Lemma 4.2 combined with (6.3b) shows that, for every numerical polynomial P on a, (2) R'pz>Dp(Rfp) and R'p is embedded into R p -®z p Z p . Numbering the elements in I, writing the i-th conjugate of a e F as cc^ and replacing t a by exp(-EiOC^Vi), we see that (3)

fv&x(t) corresponds to £(x)G(y,A,x,%) (see (2.4.3)),

where Li(y) = IjVi (j) yj, %i = £(vO and x = (xi) given by x = EiXiVi. Finally note that the norm map N : a —> Z is a numerical polynomial on a and

3.8. p-adic Shintani L-functions of totally real fields

101

hence we have an invariant differential operator D# e 2XT/z) (D^(ta) = N(a)ta). Then, directly from (2.4.8c), we get Proposition 1. For each £ e Hom(a,C x ) satisfying \ £(v01 < 1 and ^(vi) & 1 for all vi in v, we have (-n)l,A,x,x) = (DN)n(fv,^x) I t=i for all n e N. Write ^ = {UvGvC(v)}ria. Then a+ = U e eE£^i (disjoint union). For any function <|) on a having values in C, we put

Let <|) : a/fa —» C be a function for an ideal f of O. Then c|) is a linear combination of additive characters \\f e Hom(a//a,Cx). In fact, for any given a G a/Zfl, we have

by the orthogonality relation (Lemma 2.3.1). Since $ =A^(a)"1Z\j^>(a)xa> $ is a linear combination of additive characters \|/. Write ([) = Z\|^(t)V and put

This is legitimate because I ^(VJ) | < 1 . Then, noting

\|/!; e Hom(a,C x ) with

| \|/^(vi) | < 1 if

we have with the notation of Theorem 7.1 has an analytic continuation to the whole complex s-plane and = U^(-n) for all n e N and satisfies ( D ^ f f ^ l ) = (DN)nWf^(l) for each function <|>: aJCa-^ C. Now we fix a prime ideal f of F with O/C= Z//Z for a prime / ^ p. The additive group alia is a cyclic group of order /. We consider the function <>| : fl//a-> C given by (|)(x) = X\|/*idYM, where \|/ runs over all non-trivial additive characters \\f : alta-* Q x . By the orthogonality relation (Lemma 2.3.1), we see that

102

(5a)

3: p-adic Hecke L-functions if

<>(x) = -J

x e aC,

Then (5b)

ffl,(J)(t) = "X\)/^idXv€ vXxe R(v)fv'V.x(O»

where \|/ runs over all non-trivial characters \|/ : alia-* Q x . We assume that v\€ at for all v = (vi) e V and for all i. This is possible because there are only finitely many ideals which violate this condition (such an ideal is a factor of via'1 for some i). Then by (lc), f^ e R'p. Thus we have from Theorem 7.1 Theorem 1. Let the notation and the assumption be as above. Then there exists a p-adic measure \xa,ce Meas(ap;Zp[\ii\) such that for all locally constant functions f : Op = lim alpaa -> Q and n e N,

Jf(x)N(x)ndMx) = Uf*(-n), [ if

x £

if x

G

at,

a£

§3.9. p-adic Hecke L-functions of totally real fields We consider the ray class group ClF(pa) defined in §1.2. If a > (3 > 0, we see from the definition that 2+(p^) 2 3?+(pa) and thus we have a natural map Cl F (p a ) = I(p)/2V(pa) -» /(p)/^P+(pP) = Cl F (p p ), where /(p) = {^ = "7 I » and ^ are integral and prime to p}, 2>+(pa) = lP + n{a0 | a e F X , a = 1 mod x p a } (see Exercise 1.2.1). Thus we may consider the protective limit ClF(p°°) = Mm Cl F (p a ), a which is a compact group. By definition, we have a natural group homomorphism (la)

ia : (O/paO)x -> Cl F (p a ) given by a mod p a h-> the class of aO. Here we have implicitly assumed that a is totally positive. Let E be the subgroup of totally positive units in O*. Then it is obvious that Ker(ia) = the image of E in (O/paO)x. The cokernel of i a is just C1F(1) = HT+. Taking the projective limit, we have an exact sequence

3.9. p-adic HeckeL-functions of totally real fields (lb)

103

1 -> E -* O p x -> ClF(p°°) -> Clp(l) -> 1,

where 0 P X = lim (cyp a 0) x and E = lim Ker(ia) is the closure of E in OpX. Note that Op = Ylp\pOp for prime ideals p and as seen in §3, we have for sufficiently large a that log : l + p a 0 p - > Op induces an isomorphism from the multiplicative group l+p a 0 p to an open additive subgroup (of finite index) in Op. Since rankz p 0 p = [F:Q], we have Opx = |ixZp [F:Q] as topological groups with a finite group \i. Since E is a Zp-submodule of OpX, it is of finite rank over Z p . Since rankzE = [F:Q]-1 by Dirichlet's unit theorem (Theorem 1.2.3), we know that rank Zp E < [F:Q]-1, i.e. O p x /E = j i ' x Z p 1 + 5 for a non-negative integer 8 and a finite group JLL*. Although the equality of the ranks, rankz p E = rankzE, is conjectured by Leopoldt, this is still an open question except when F/Q is an abelian extension (a theorem of Brumer confirms this conjecture when F/Q is abelian [Wa, Th.5.25]). Anyway, G = C1F(P~) is a compact group, and we can decompose G = GtorxW non-canonically so that G tor is a finite subgroup and W = Z p 1+5 . The group /(p) of fractional ideals prime to p can be naturally regarded as a dense subgroup of G. On /(p), we have the norm map N : /(p) -> Z(p) = Z p HQ. This map N coincides with the usual norm map on (P+ and hence a polynomial map. Thus Af: /(p) —» Z p is continuous with respect to the topology induced from G. Thus N extends to a continuous character N : G -» Z p x . Now take the completion Q of Q p and denote by A the p-adic integer ring of Q. (Note that Q is substantially larger than Q p ; see [BGR, 3.4.3].) The space C(G;A) of all continuous functions on G with values in A is naturally equipped with the uniform norm I f I p = SupxeG I f(x) I p . Let Meas(G\A) denote the space of bounded p-adic measures on G with values in A: ) = HomA((:(G;A),A). We can define the uniform norm on Meas(G\ A) by

|(p| p = Sup| f | = ilJ G fd(p|p.

104

3: p-adic Hecke L-functions

Theorem 1. For each element a e G = C1F(P°°), there exists a unique p-adic measure £ a on G such that for any character % : O F C P " ) -> Q x and any n G N, we have JGX(x)iV(x)ndCa(x) = (l-X(a)^V(a)n+1)np|p(l-Xo(pMf)n)^(-n,%o)^ where we denote by L(s,%) the primitive L-function (i.e. the L-function of primitive character %Q associated with %) and hence the factor (l-%0(p)N(p)n) is non-trivial if the primitive character % has conductor prime to p. Proof. For each ray class c in C1F(1), we choose a representative a- c^ which is prime to p. Then we choose V as in the previous section so that v is contained in a for all v e V . We then choose a prime ideal C with a/aC= Z//Z for a prime / in Z such that vi £ at for all i and all v G V. Among the prime ideals C with alia = Z//Z for a prime / in Z, there are only finitely many which do not satisfy the above condition for a fixed V. Thus by changing I if necessary, we have the measure (i^/ on av= Op as in Theorem 8.1 for all c e Clp(l). Since G = Uc^c^C^p^E) (disjoint union), to each function <|) G C(G;A), we can associate a continuous function <|>c (c G Clp(l)) on Op by17 < M x ) = { ^ " l x ) * X G °P X ' Then we define 10 otherwise.

This certainly defines a measure on G. We now compute (2) By Theorem 8.1, we see that

since a+ = UzeE£%a and the integrand is invariant under multiplication by E. Thus l

J

X a

)N(aa ^

3.9. p-adic HeckeL-functions of totally real fields

105

where oca * runs over all integral ideals prime to p which are in the same class as a"1 in Clp(l). This combined with (2) shows the desired formula when a = L The uniqueness follows from the fact that the polynomial functions and locally constant functions are dense in 0(G;A). Now the ideals i for which the measures K^t are constructed are dense in G by Chebotarev's density theorem (see §1.2). Thus for general a e G, we can choose a sequence {4} of such ideals converging to a. Then by the evaluation formula, £n = C^ converges to £a in Meas(G\A), which finishes the proof. Using the notation of §5, we now define the p-adic Hecke L-function with character % : ClF(pa) -> Q x by £P(s,X) =LF,p(s,%) = (l-%(a)(iV(a))1-s)-1JG%(x)co-1(iV(x)>(iV(x))-sdCa(x), where co is the Teichmuller character of Z p x . Then Lp(s,%) is a p-adic analytic function on Z p except when % = id, and in this special case, £ p (s) = £p,p(s) = Lp,p(s,id) is a p-adic meromorphic function defined on Z p -{1} and having at most a simple pole at s = 1. Moreover we have the following evaluation formula: "^o) for all m e N, where we write simply co for co°iV, and the right-hand side is the value of the complex L-function while the left-hand side is the value of the p-adic L-function and the values are equal in Q. The value of the p-adic L-function at positive integers are unknown except the value at 1 for F abelian over Q (see §5). When F is abelian over Q, CF,P(S) = Ilx^Q,p(s,%) by class field theory for Dirichlet characters associated with the field F. (For class field theory, we refer to [N] and [Wl].) Thus we know the residue at s = 1 of £p,p by the result in §5. In fact, for general F not necessarily abelian, Colmez [Co] proved the following p-adic residue formula: (4)

Ress=iCF,P(s) =

where h(F) = (/:£) is the class number of F, w is the number of roots of unity in O (thus w = 2) and choosing a basis {£i,...,e r } of the unit group O*, Rp = ±det(/og(ei^))ij=i r_i and Dp is the discriminant of F. Here e® is the j-th conjugate of e in Q p and "log" is the p-adic logarithm defined in a "P

neighborhood of 1 in Q p . For the choice of the sign of -T-^- and the proof of

106

3: p-adic Hecke L-functions

the formula, see [Co]. The Leopoldt conjecture for F and p is equivalent to the non-vanishing of Rp. This formula is a generalization of Theorem 5.2 and is an interesting p-adic analogue of Theorem 2.6.2 and Corollary 8.6.2. Although we have only discussed p-adic abelian L-functions over totally real fields, we can also construct abelian p-adic L-functions over CM fields. We refer to [K3], [K5], [dS] and [HT2] for the various constructions of such p-adic Hecke L-functions. The existence of abelian p-adic L-functions is the starting point of Iwasawa theory, which studies a subtle but deep interaction between such p-adic L-functions and the arithmetic of abelian extensions of the base field F. We refer to [Wa], [L] and [dS] for basics of Iwasawa theory. The so-called "main conjectures" in the Iwasawa theory have been proven recently in many instances. Although there are no books written on this subject yet, we refer for recent developments to the following research articles: [MW], [Wi2], [MT], [R] and [HT1-3].

Chapter 4. Homological Interpretation In this chapter, we will give a homological interpretation of the theory of the special values of Dirichlet L-functions over Q and will reconstruct p-adic Dirichlet L-functions by a homological method (called the "modular symbol" method). A similar theory might exist for arbitrary fields, but here we restrict ourselves to Q. The modular symbol method was introduced by Mazur [MzS] in the context of modular forms on GL(2) as we will construct later, in §6.5, p-adic L-functions of modular forms (on GL(2)) by his original method. Basic facts from cohomology theory we use in this section are summarized in Appendix at the end of this book. We use standard notations for cohomology groups introduced in Appendix without further warning and quote, for example, Theorem 1 in the appendix as Theorem A.I. If the reader is not familiar with cohomology theory, it is better to have a look at Appendix before reading this chapter.

§4.1. Cohomology groups on G m (C) We consider the space T = C/Z, which is isomorphic to G m ( C ) via z h-> e(z) = exp(27ciz). Thus T = P ^ O - f O , ^ } , where P*(C) is the projective line. We apply the theory developed in Appendix to X = P 1 (C) and Y = T. With the notation of Proposition A.5, S is first {0,°°} and later will be M-NU{0,oo}, and So will be {0,°o}. We have 7Ci(T) = Z. Let A be any commutative algebra. Let L(n; A) for 0 < n e Z be the subspace of the polynomial ring A[X,Y] consisting of homogeneous polynomials of degree n. We let the additive group Z act on L(n;A) by vP(X,Y) = P(X-vY,Y) for v e Z. Then we define a sheaf Zln;A) = L(n;A) on T (with the notation in Theorem A.I) by the sheaf of locally constant sections of the projection (CxL(n; A))/Z —> T, where n e Z acts on CxL(n;A) by n(z,P) = (z+n,nP). We identify C/Z with (-i©o,ioo)xR/Z and compactify it as T* = [-ioo,i«x>]xR/Z (i = >Tl). With the notation of Appendix, we have T* = T s ° for S o = {0,°o} = {±i~}. Let N be a positive integer, and take out N small open disks around -^ ( r e Z) from T*. The resulting space we write as T^ for S = N^Z/ZU {+<*>} = |LINU{0,OO}. We also consider TN~ ° (resp. T^°), removing the boundaries from T^ around s e So = {±°°} (resp. s e S-So). This notation fits well with Proposition A.5. We write the boundary around s e S of T^ as dsT^. Note that T^~S° is no longer compact. We want to study Hq(TN,£(n;A)) and the compact support cohomology group H2(TJ5["So,.£(n;A)) on T^~S°. We first compute the homology group U\(T^ ,9 Tjjf ,A) for d Tjjf = TooLJT.oo, where T+oo = ±°ox(R/Z). Let cr be a small circle centered at r e S inside Tjjf and

108

4: Homological interpretation c r be the vertical line passing through r e R . We write Coo for the circle added at ©o. Then we see easily that

fHi(Tfp,3Tgp,A) = Ac°0{0rG(Z/NZ)Acr} (la)

<

rfc^/iM^,

[Hi(T N ,A) = ACoo®{®rG(Z/NZ)Acr}. S—S

We now compute fti(TN °). Fixing a base point x, we draw the line from x to c r (r e Z or ±©°) and turn around the circle in the positive direction and return to x. This path will be denoted by 7tr. Then F = 7Ci(T^~S°) is generated by 7Cr (r = ±<*> and r G Z/NZ), and there is only one relation among them: ^o where the product is taken in the increasing order for the index 0 < r < N. By using the exponential map e(z) = exp(27iiz), we can identify TN-S with G m -n N for ^ N = ( C e C X | C N = 1 } . Identifying Gm-|iN with P^jiiN-fO,©©}, we can identify F with TCi(Gm-|iN)Anyway we have a natural action of F on L(m;A). Of course this action factors through 7Ci(T) = 7CooZ = {7Coom I m e Z}. We write F^ for the stabilizer of £ G |IN« Then F^ = K^Z (where n^ = nT for C, = e(r/N)) acts trivially on L(m; A). We have a natural map res: H ^ F ^ m j A ) ) -> H^r^LCmjA)) = H^Cr/N^CmjA)) = L(m;A), which fits into the following exact sequence (Corollary A.2): (lb)

Let

gf ,£(m;A)) . Then

because they are of the same homotopy type. Moreover TN does not have any boundary and hence H C ( T N , A ) = A. By Proposition A.6, H2(TN,£(m;A)) is dual to H C (TN,-£(HI;A)) = 0 as long as A is a Q-algebra, because there are no compactly supported (locally constant) global sections except 0. This shows H2(TN,£(m;A)) = 0. We give a different proof of this fact: We first prove H J } R ( T , C ) = 0. We need to show that for any co = fdxAdy for z = x+iy,

4.1. Cohomology groups on Gm(C)

109

co = dT] for some r\. Defining F(z) = J y f(x + it)dt, we have P- = f, and F is a function on T. This shows that d(Fdx) = { ^ d x + ^ d y } Adx = -fdxAdy. Thus d(-Fdx) = fdxAdy, which shows the result in this case. For general TN, we take a small neighborhood U of each hole x = ^ isomorphic to a punctured disk D at the origin. Then D is isomorphic to a cylinder by zh> log(z). Thus any differential 2-form co can be considered to be a 2-form on a cylinder and therefore is of the form drj. Thus by pulling rj back to U, locally any differential 2 form co is equal to drj on U for some rj. Taking a C°°-function on T which is equal to 1 on a smaller open disk in U and vanishes outside U, we know that the support of co-dr| does not meet the hole. Thus by changing a representative co of the cohomology class in H J ) R ( T N , C ) , we may assume that co is well defined on T (i.e. CO does not have singularities at the holes). Then as already seen, co = dr| and hence H Q R ( T N , C ) = {0}. Since C is faithfully flat over Q, we see that H2(TN;Q)QC = H 2 (T N ,C) = {0}. Thus H 2 (T N ,Q) =0, and hence we see that H2(TN,A) = H 2 (T N ,Q)® Q A = {0}. Now we prove H2(TN,£(m,A)) = {0} by induction on m. We consider the exact sequence of F-modules 0 -> L(m;A) x Y > L(m+1;A) —2-» A -> 0, where 7i(P(X,Y)) = P(l,0). This induces an exact sequence of sheaves (*) 0 -> X(m; A) x Y ) £(m+l;A) —^-»A -> 0 from which we get another exact sequence 0 = H2(TN,Am;A)) — ^ U H 2 (T N ,Am+l;A)) ^ U H2(TN,A) = 0. The vanishing of H2(TN,£(m;A)) follows from the induction assumption. This shows H 2 (T N ,£(m+l;A)) = 0. We see easily that H°(TN,iXm;A)) = L(m;A) r = AYm. Thus we have from (*) an exact sequence 0 -> A -> HKTN^mjA))

xY

) H ^ N ^ m + ^ A ) ) — ^ H ^ T N . A ) -> 0.

Thus if A is a Q-algebra, dimAH1(TN,A)+dimAH1(TN,i:(m;A))-l = Since dimAH^TNA) = N+l, we see that dimAH1(TN,iXm;A))+N = In particular, we get (2)

^

110

4: Homological interpretation

Each inhomogeneous 1-cocycle u of F is determined by its values u(7C^) for £ E JINU{°°} because of the relation: 7CooIIrG (Z/NZ)71*71-00 = *• Thus w e c a n embed the module of 1-cocycle Zl(T,h(m;A)) into copies of L(m; A) res : Z 1 (r,L(m;A)) <-> L(m;A)[Z/NZ]0L(m;A) where L(m; A)[Z/NZ] denotes the module of a formal linear combination of elements in Z/NZ with coefficients in L(m;A), which is in turn isomorphic to L(m;A[Z/NZ]) for the group algebra A[Z/NZ]. On the other hand, since K^ for £ e |IN acts trivially on L(m;A), res brings the submodule B^FJLOnjA)) of coboundaries into (7Coo-l)L(m;A) resCB^JLteA))) = (7ioo-l)L(m;A). Thus we have (3a) rfCTttAmjA)) = tf^imA) = L(m;A)[Z/NZ]eL(m;A)/(7Coo-l)L(m;A). We define Hp (T^,L(m; A)) by the kernel of the natural restriction map (3b)

res : H\TN,L(™;A))

SH1^

0

,£(m;A)) -> H

1

^

0

,£(m;A)).

Then we see from the relation that 7Toorire (Z/NZ)7^71-00 = ^ (3c)

Hj>(TN,£(m;A)) = {u e L(m;A[Z/NZ]) | {u} G (7Coo-l)L(m;A)},

where {SrGZ/NZUrr} = SrGz/NZUr, and from (2) (3d)

rankAHp(TN,Z<m;A)) = N(m+1)-1, if A is a Q-algebra.

We now define Hecke operators T(n) for each integer n & 0 acting on the cohomology groups. We consider the projection map % : C/nZ —> C/Z. We put V = TC"1(TN) and 7ti(V) = O as a subgroup of F. Since the projection map % : V —» TN is a local isomorphism, we have two natural maps: 7t* : H ! (T N ,i:(m;Q)) -> H ^ V . ^ t o Q ) ) and Tr : H ^ V ^ m j Q ) ) -> H ^ T ^ ^ m j Q ) ) . The existence of the morphism TC* is obvious. We explain the construction of the trace operator Tr. Since the projection n : V —> TN is a local homeomorphism, for each small open set U in a simply connected open set in TN, ^ ( U ) is isomorphic to a disjoint union of open sets each isomorphic to U. Write simply M = L(m;A). We write n* for the direct image functor, i.e. 7i*M is the sheaf on TN generated by the presheaf U f-> M(n'l(\J)). We take an open subset Uo in rc^CU) so that K induces Uo = U. Then by definition, we know that

4.1. Cohomology groups on Gm(C)

111

= M(Uo)d for the degree d of K. This isomorphism is explicitly given as follows. We may identify rc'^U) with the image of the disjoint union Ui8i(Uo) in V, where {5i} is a complete representative set for O\F. Here we have a commutative diagram for the universal covering H of TN: n H ° ) C - N ^ Z / n Z — ^ L - * V —2-» T N

l&{

i&{

H

> C-N^Z/nZ

I > V

in which Si: H = H naturally induces Si: C X - | I N = C X -JIN- Then we identify M(5i(Uo)) = M with M(Uo) = M via the map: M(5i(Uo)) 9 X H Si^x e M(Uo) = M(U). Now it is clear that 7C*M/s = Indr/<j>(M) on TN, where Indryo(M) is the induced module M®z[]Z[r] and the F-action is given by y(m<8>a) = m^ay"1. Note that the direct image of a flabby sheaf is flabby by definition and that 7t* is an exact functor by (A.5a), because n is a local homeomorphism. Therefore any flabby resolution of M/s gives rise to a flabby resolution of

(TC*M)/TN

Just ^v

applying re*. Thus we know that J * M )

= H^(V,M) (Shapiro's lemma).

Now we define Tr : 7t*M/rN(U) -> M ( U ) / T N by Tr(x) = ZiSi^x. This induces a morphism of sheaves. Obviously this is induced from the trace map of algebras Tr : Z[T] -> Z[O] id<8>Tr : Ind r /o(M) = M<8>z[ M. Then Tr induces a map of cohomology groups Tr : HUV,M) = ^ ^ (I

For a n = L

0\

L we define a n (z) = z/n. Then a n induces a morphism of

sheaves ocn : £(m;A) / v -» X(m;A)/TN by (z,P(X,Y)) H> {zln?((X,Y)lan)), which in turn induces a morphism of sheaves ocn*£(m;A) -> X(m;A)/v, where the inverse image an*^C(m;A) is the sheaf on V generated by the presheaf U \-> X(m;A)(On(U)). Then we have a natural pull back map a n * : rf(TN,£(m;A)) -> ff(V,jC(m;A)). Then we put (4) T(n) = Troa n *. Note that the action of oc^ preserves boundaries at ±°o. Therefore we can define the operator T(n) in the same manner on H'CT+oc^mjA)), HJ(3 T^0,L(m;A))

112

4: Homological interpretation i

S—s

and H C (T N 0,£(m;A)). Then the boundary exact sequence is compatible with the action of T(n). More generally, we can let an upper triangular matrix fa b\ a = e M 2(Z) (ad * 0) act on C and L(m; A) by <x(z) = (az+b)/d and aP(X,Y) = P((X,Y)V), where a 1 = det(a)a" 1 . We insert here a computation of the Fourier transform on finite abelian groups which will be used later. Let <|) : Z/NZ —> C be a function. We define the Fourier transform <j) = f(§) : Z/NZ -> C by

Then we see from Lemma 2.3.1 that, for J() | t(x) =

(5a)

I -Y

IZ^ A M Y e(x(ty"z)^ - IN*(z/0 e

-2,yGz/NZ^W2.xEZ/NZ ^

N

' ~ jo

if

tl2*

otherwise.

Now we apply this Fourier transform to a primitive Dirichlet character % : (Z/CZ) X -> C x for a divisor C * 1 of N. We write N = N'C. We extend the character % to a function % : (Z/CZ) —> C just defining its value to be 0 outside (Z/CZ)X. Then we compute (5b)

mod N 1

f—^ - IZ" (x/N')G(x)N' if N ' | x , ( N l ) " {A 0 otherwise. In other words, %(tx) with 0 < 11 N1 is supported on (N7t)(Z/NZ)x. We now compute the action of T(n) on H ^ T N ^ ^ C ) ) using differential forms (see Theorem A.2). We consider differential forms with values in JJ(m;C) of the following type. For an integer 0 < j < m (6a)

O)j(f) = f(z)(X-zY)m~jYjdz for any meromorphic function f on T.

In particular, we study the following explicitly defined meromorphic functions. Let |LL: Z -> {±1, 0} be the Mobius function and % be a primitive Dirichlet character modulo C. Then we put

4.1. Cohomology groups on G m (C)

(6b)

113

f(z) = fid(z) = 0 ^ , f.,id(z) = Xo
f>< nz ) for 0< s I N, n=l,(n,s)=l

(6c)

fx(z) = X~=0X(n)e(nz)

=

X(-l)c'iyL2=ie^

C-l

C-l

j=0

j=0

n

Since FooanFoo = (J _ 0 Foo(xnj for a n ,i = L

I, it is easy to see from defini-

tion that the action co i-> Zj(oCn,j*co) I cxnj on differential forms induces the operator T(n) on de Rham cohomology groups with coefficients in L(m;A) (see Theorem A.2), where co I a n j(x) = anjlco(x) (a 1 = det(a)a- 1 ) for the action of a n j l on L(m;C) and oCnj*co is the pullback of co under the action of a n j on C. We now compute the action of the operator T(n) acting on C0j(fx): o)j(g) | T(n) = ^ a ^ C D j t g ) I an>i i-l V ^ n - 1 . - ,

nJ

x+lY

. - r Z+i ^rvm-i /Z+i N ,

nY)

(

)dz

, i-1

ffl nJ

1

+L

= Ei( -ir* g ir = < Then it is easy to see that C0j(g) | T(n) = ©j(g | jT(n)) for g | jT(n) given by (7a)

gljT(n)=nJ-1 X": o 1 g(^)-

If g has a Fourier expansion g(z) = £°°_ a(m,g)e(mz) (a(m,g) e C), then we see that (7b)

|

J

Since C0j(fx) gives a closed form, we have its de Rham cohomology class [cOj(f%)] in Hb R (T N ,Am;C)) (see Theorem A.2). Then we have [co/g)] | T(n) = [coj(g) | T(n)] = [cOj(g I /T(n))]. We compute the action of T(n) on fx using the formula (7b). The functions in (6b) have two Fourier expansions: one at °° and the other at -°°, which are given as follows. Since f(z) = , \ \ , we may expand it into the geometric series f(z) = ^ ^ e O n z ) if Im(z) > 0. Writing the same function as f(z) = , , , , we have the expansion valid on the lower half plane: f(z) = -l-£" = 1 e(-nz) if Im(z) < 0. We now list the two expansions of f% for % * id, which are computed using the above expansion of f and the definition of f%:

114

(8a)

4: Homological interpretation

f*(z) = i

S~ X(n)e(nz) if Im(z) > 0, n 2 [-S n=1 %(-n)e(-nz) if Im(z) < 0.

We define g | t(z) = g(tz) for each divisor t of N. Since the Fourier expansion of fx with % T* id and f I t-f with 1 < 11N has no constant terms, the cohomology classes [C0j(fx|t)] and [cOj(f-f|t)] actually fall in the parabolic cohomology group; that is, (8b)

[coj(f% 11)] G H ^ T N ^ i r ^ C ) ) for % * id and 0 < 11 N \ [C0j(f-f|t)]€ Hj>(TN,i:(m;C)) for K t | N .

By (7b), we have (8c) C0j(f% 11) | T(n) = nj%(n)o)j(fx 11), co 0 (l) |T(n) = coo(l) for n prime to N.

Theorem 1. The set of cohomology classes {[a>j(fx 11)] | 0 < j < m, 0 < 11 N'}U{[co0(l)]} (= {[(X-zY)mdz]}) forms a basis of H^T^iXn^C)), where % runs over primitive characters mod C including the trivial character mod I, and we have written N = N'C. Proof. Let Q x = {[©j(fx 11)] | 0 < j < m, 0 < 11 N f }. Writing W x for the subspace of H^TN.XOIIJC)) spanned by Q,%, we see from (8c) that W^flZ^W^ = {0}. When % & id, f% \ t has non-trivial simple poles at P t = ( x e N"XZ | Ctx G (Z/CZ)X) by definition, where C is the conductor of %. Here we understand C = 1 for % = id and (Z/1Z)X = {0}. This shows JCx©j(fx 11) is non-zero if and only if x G P t . Therefore Q% is linearly independent in W x . Thus writing the number of positive divisors of N as d(N), we see that d i m c W x = d(N')(m+l). Moreover by computation, we know that JCeoCGo(l) e (7ioo-l)L(m;C) (see (9b) below) and hence coo(l) e ZXWX. Since we know that

dimcH 1 (TN,^(m;C))

by (2), what we need to show is

5LdimcW x = N(m+1). Writing (ppr(C) for the number of primitive characters modulo C, we have an obvious identity: £o
JCxO)j(fx) = -

4.1. Cohomology groups on Gm(C)

115

Similarly the value of JZ coo(l) at (X,Y) = (1,0) is (J z Z + 1 coo(l))(l,0)=L

(9b) 1

fZ+l

Noting that Hf)R(Coo,M) = M/(7Coo-l)M via (9b) shows that the cocycle yi-» J

CGH>

Jco (1,0) for M = L(m;C),

coo(l) is rational. Thus we have

Corollary 1. Let K be the field generated over Q by the values of all characters of (Z/NZ)X. Then the following elements form a basis of H1(TN,£(m;K)): {G(%"1)o)j(fx | t)}j>t,xU{coo(l)} in the notation of Theorem 1. Moreover we have for primitive %. N m-J G ( % -i ) 0 ) j ( f x ) G n\TN,L(m;Z[xl)) Since T^ (C, = eta)) acts trivially on L(m;A), we have a natural restriction map res r : H ^ T N , L(m;A)) -> H

1

^ , L(m;A)) s L(m;A).

For each closed form co, this map is realized by coh->Jc co. Then we define

(f>m = q>: tfCTN.XdmA)) -> L(m;A[Z/NZ]) by q>(x) = x T l " f ]res r (x)r. r=ov

u

x

y

By definition (p isinjectiveon Hp(TN, ^(m;A)) and

T(n)(cr/N) = X c(r/Nn)+(i/n)i=l

Note that C(r/Nn)+(i/n) = 0 in the homology group except when r+Ni = 0 mod n. If r+Ni = 0 mod n, C(r+Ni)/Nn = cn-ir/N for n-1r G Z/NZ. This shows (10)

T(n)(c r/N ) = cn-ir/N.

Note that H1(TN,£(m;A))/Ker((pm) = {H1(TN,A)/Ker(q>o)}®L(m;A), because n^ acts trivially on L(m;A). This isomorphism at the level of differential forms is

116

4: Homological interpretation

given by O)j(f) H> fdz®X m " j Y j = f

^ O)j(f)(z). Then it is clear that when

A = C, the action of T(n) is interpreted on the right-hand side of the above identity by T(n)®a n for T(n) on H ^ T ^ C ) . Then the above formula (10) shows the desired result for A = C. The result for general Q-algebras A then follows from the result over C because H1(TN,X(ni;A))(8)AC = Kl Let Hm(N;A) be the A-subalgebra of EndA^HT^AmjA))) generated by T(n) for all n prime to N. For any A-algebra homomorphism X : Hm(N;A) —» C, we define H 1 P (T N ^(m;A))[?i] = {x e H ^ T N ^ ^ A ) ) | X | h = X(h)x}. All the X's are explicitly given by ^(T(n)) = n}%(n) for Dirichlet characters % : (Z/NZ)X -» A x and integers 0 < j < m. We call X primitive if the associated Dirichlet character is primitive modulo N. Then we get from Proposition 1 Corollary 2. (i) H m (N;A) is isomorphic to the A-subalgebra in EndA(L(m;A))<S>A[Z/NZ] generated over A by an®(nl mod N) for all n prime to N. In particular, H0(N;A)= A[(Z/NZ)X]. We now suppose that A is a Q-algebra. Then we have (ii) Hp(TN,X(m;A))[A,]=L(m;A[Z/NZ])[A,] is free of rank one over A if X is primitive; (Hi) Suppose N is a prime. Then Hp(TN,X(m;A))[X] is free of rank one over A if A,(T(n)) = 1 for all n prime to N; (iv) Hj,(TN,X(m;C))[X] = C(Oj(f%) if X is primitive and ?i(T(n)) = nj%(n) for all n prime to N; (v) Suppose N is a prime. Then Hp(TN,£(m;C))[Aj = C(Do(f-f I N) if = 1 for all n prime to N. Now we consider the natural projection map p : H ^ T ^ ^ m j A ) ) ^ nl?(TN,L(m;A)). We analyze Ker(p) as a Hecke module. By the boundary exact sequence (lb), Ker(p) is the image of H°(3Tgp ,£(m;A)) = H0(Too,£(m;A))eH°(T.eo,i:(m;A)) s H°(R/Z,X(m;A))2. We see easily that H°(R/Z,£,(m; A)) = H°(Z,£(m; A)) = AYm if A is a ring of characteristic 0. Then it is easy to see from the definition of T(n) that

Y m | T(n) = Xr="o a n,i Ym = n m + 1 Y m . Thus T(n) acts on the one dimensional space Ker(p) via the multiplication by n m+1 . The eigenvalue n m+1 of T(n) does not appear in Im(p). Thus we have ^ ^ S ^ ( m ; A ) ) s Im(T(n)-n m+1 )eKer(p)

4.2. Cohomological interpretation of Dirichlet L-values

117

as long as A is a Q-algebra, and therefore we have a unique section (11)

i : 4(T N ) X(m;A)) -> H ^ T ^ .

satisfying loT(n) = T(n)oi for all n. Since [Gft'^N^cOj^)] for a primitive % is an integral element which is an eigenform of all T(n) and is cohomologous to a compactly supported form, we see that i([G(x-1)NJ©j(fx)]) G Since T(a)-am+1 is well defined over H*(TN~So,i:(in;Z[%])), we have for any integer a > 1 prime to N (12) (T(a)-am+1)i bring the image of HJ>(TN,4m;A)) in H1p(TN,iXm;AzQ)) into the image of Hc(TN~s°,£(m;A)) in H*(TN~s°,£(m;A®Q)) for any sub algebra A in C.

§4.2.

Cohomological interpretation of Dirichlet L-values

We fix a primitive character % * id mod. N. We write X : Hm(N;Z[%]) -> Z[%] for the Z[%]-algebra homomorphism given by X(T(n)) = %(n). We fill the hole at 0 of T N = T-N"1Z (resp. T^°) and call the new space YN (resp. X N ). By (1.10), T(n) for n prime to N still acts on H ^ Y N , £(m;A)). Since ©j(f%) does not have a pole at 0, we may consider [co/f^)] e Hp(YN,£(m;C)). Let 8% = G(x"1)[co0(fx)]. By (1.9a) combined with Corollary 1.2, Nm8% generates the A,-eigenspace Hp(YN, L(m;A))[X] defined in Corollary 1.2. We consider the integral J 08% on the vertical line c° passing the origin. By (1.8a), we have m

(D

Jc
The embedding R —> YN given by y i-» iy induces a morphism Int : E£(Y N , £(m;A))-» H^(R,£(m;A)) =L(m;A) (co h-> f°° ©). Here X(m;A) on R is a constant sheaf. By Corollary 1.1, for each integer a prime to N, the cohomology class (am+1-%(a))[NmG(%"1)coo(fx)] for primitive % is integral (i.e. is contained in the image of H X (XN, £(m;Z[%]))); moreover, by (1.12), it can be regarded as being contained inside the image of Hc(X N , L(m;Z[%])) in HC(XN, X(m;Q[%])). Then by a theorem of de Rham (Theorem A.2), we see that

118

4: Homological interpretation

(a m+1 -%(a))N m J c0 8 x = Int(N m (a m+1 -x(a))8 x ) To include the identity character % = id, we modify a little the cycle c°. Instead of c°, we take a small real number e > 0 and put c° = c e . Since c° and c° are homologous in

HI(XN,3XN;Z),

we get the same result: Joco = Jceco as long

as 0) is holomorphic at 0. We define Int' on Hc(TN~S°,£(ni;A)) using c° in place of c° in the same manner as Int. The map p m : L(m;A) —» YL(m-l;A) given by pmdiaiX^Y 1 ) = I^iaiX" 1 "^ combined with Int' gives Int m : H ^ Y N , L(m;A)) -> H*(R, YL(m-l;A)) = YL(m-l;A). The power z1 of z of the coefficient of X^Y 1 in (X-zY)m kills the pole at z = 0 if i > 0, and thus we can compute the map Intm in the same way as Int even for coo(f-f'q). Then for a prime to q, (am+1-T(a))Intm(qmco0(f-f|q)) (

^

j

)

1

j

J

e YL(m-l;Z).

Thus this proves again the result we obtained as Corollary 2.3.2: Theorem 1. Let % ^ id be a primitive Dirichlet character modulo N and a be any integer prime to N. Then J!NjG(x-1)(27iO;j-1(l-%(-l)(-l)j)(am+1-%(a))L(l+j,x)G Z[%] and (27cO" j ' 1 (l-(-l) j )(a j+1 -l)q j (l< j - 1 )j!C(l+j)e Z for all primes q. By the functional equation, we have for j e N, (2)

L ( - j , x -

1

)

j

J

1

j

1

§4.3. p-adic measures and locally constant functions In §3.3, we studied the structure of the space of p-adic measures on Z p in terms of interpolation series. Here we describe the space via locally constant functions. Let p be a prime and G be a topological group of the form G = |UxZpr for a finite group (I. We put Gi = (p^p) 1 . We fix a finite extension K/Q p and write A for its p-adic integer ring. We equip K a normalized p-adic norm | | p such that I p I p = p"1. For any topological space X, we write LC(G;X) for the space of locally constant functions on G with values in X. Thus a function (|) : G —> X is in LC(G;X) if and only if for any point ge G, there exists an open neighborhood Vg of g in G such that the restriction of <>| to V is a constant function. By definition, it is obvious that for any locally constant function § and for any subset S of X, §A(S) = Ug^-i^Vg is open; in particular, <]) is

4.3. p-adic measures and locally constant functions

119

continuous. Since G is compact, G = UgGGVg implies that we can find finitely many points gi, ..., gs on G such that G = UjS=1Vg.. By the definition of the topology of G, a basis of open sets of G is given by {g+Gji g e G, i = 0,l,---}. Thus for large i, Vg- z> gj+Gi, that is, $ induces a function §i : G/Gi —» X and <> | = <J>iO7Ci for the projection %[: G —> G/Gi. The space £(G/Gi;X) is made of all functions on the finite group G/Gi with values in X and is isomorphic to the set X[G/Gi] of formal linear combinations S g G G / G i x g g with x g e X via <>| »-> SgeG/Gi<|)(g)g. Thus we see that (1)

LC(G;X) = lim C(G/G{;X) = lim X[G/Gi]. i

i

For a topological ring R, we define the space of distributions 2fo<(G;R) by (2)

0ist(G;R)

= Hom R (£C(G;R), R).

If cp e £fo t(G;R) and if Xs is the characteristic function of an open set S of G, we write (p(S) for cp(xs)- Since %h+Gi = SgGGi/Gj%h+g+Gj for j > i , we have the following distribution relation:

(3)

(p(h+Gi) = XgeGi/c/Pdi+g+Gj) for all h e G and j > i.

On the other hand, if we are given a system cp assigning a value cp(g+G0 e R for all g e G/Gi and for all i sufficiently large satisfying (3), we can extend cp to a distribution as follows. For a given 0 e £O(G;R), taking sufficiently large i so that cp(g+Gi) is well defined and <>| = <|)iO7q with <|>i: G/Gi —> R» we define cpC) = SgGG/Gi(|)i(g)cp(g+Gi). This is well defined because for any other choice j > i of the index, we have

because 4>jCs) = iCg) for all g € G. Thus we have

Proposition 1. Let R be a topological ring. Then a function (p : {g+Gi I i > M, and g e G ) - > R is induced from a distribution if and only if cp satisfies (3) for all j > i > M. Let R be a closed subring of K. For any measure cp e Meas(G;R)y 9 induces a distribution, again denoted by cp, by (p((|)) = j(|>d(p. Then | (p(<|)) | p < |

t I p is finite. Now we want to show the converse. For any continuous function (J) : G —» R, we can find for each

120

4: Homological interpretation

positive 8 > 0 and g e G a small open neighborhood Vg of g such that I g+Gi for all g e Vg-. Then choosing a complete representative set R for G/Gi and defining <])e : G/Gi -> R by (|)e(h) = (|)(g) if h € g+Gi, we see that <|)e e LC(G;R) and | <|>e-<|> | p < e. Thus LC(G;R) is dense in 0(G;R) and (4)

I <|>e-(])e I p < I (e-<|))+(4)-e- I p» I ^"^e I p) < m a x ( e , e ' ) .

Let cp is a distribution with bounded norm I (p I p. This is equivalent to saying that | M and all g e G. Then (4) implies

I e)-£') I p ^ I 9 I P I e-0e I P < I 9 I p m a x ( e , e ' ) and {(p((|)i/n)} is a Cauchy sequence in R. We then define (p(())) = lim i/n) e R. n—>«»

Then it is easy to verify that cp e Meas(G\R). Thus we have Proposition 2. For any closed subring R of K, LC(G\R) is dense in C(G;R). Any bounded distribution on G with values in R can be uniquely extended to a bounded measure with values in R. In particular, Meas(G\A) =
§4.4. Another construction of p-adic Dirichlet L-functions We reconstruct the p-adic measure which interpolates the values of Dirichlet L-functions via cohomology theory. This type of formalism (the formalism of modular symbols) was found by Mazur in [Mzl] and [MzS] where he applied it to L-functions of elliptic modular forms (see §6.5). We fix a prime p. Let K/Q p be a finite extension and A be the p-adic integer ring of K. Let N > 1 be a positive integer prime to the fixed prime p. Let XN be the space obtained from TN° by filling the hole around 0. The inclusion: c r —» TN° induces an A-linear map: Hlc(X^,L(m;A)) —» L(m;A), which we write as \ \-> J r ^. Then we consider a map (1)

c : p - Z = UT ^

-» Hom A (H'(X N ,£(m;A)), L(m;K))

given by

4.4. Another construction of p-adic Dirichlet L-functions For £, G Hj.(XN,£(m;K)), we write c^(r) = L tiplicative semi-group

f^.

121

Here we let the mul-

M2(Z)riGL2(Q) act on L(m;K) by aP(X,Y) =

P((X,Y)V) (a G M 2 (Z)), where

a 1 = det(a)a - 1 . Then c§(r+l) = c^(r) by

definition, and c^ factors through Qp/Zp = p"°°Z/Z. Supposing 2; | T(p) = ap^ with I ap I p = 1, we define a distribution <X>£ on Z p x by (2)

O^(z+p m Z p ) = ap-ml PQ

J c ^ ( ^ r ) for z = 1, 2, ... prime to p.

This is well defined because c^(r+l) = c^(r). We take G = Z p x and fix an isomorphism G = |ixZ p for a finite group [i. Then |U = {C, e Z p x | ^ 9 ( p ) = 1 } , where cp is the Euler function and p = 4 or p according as p = 2 or not. Then the subgroup Gj = l+p x Z p corresponds to p x Z p . To show that O^ actually gives a distribution, we need to check the distribution relation (3.3). We compute p

p l (j+x) ° ) v f °]( " I c (£*\-Y o I L ^ P > ^ J l o IJ o

-xVl This shows The general distribution relation (3.3) then follows from the iteration of this relation. By a similar argument, we see that (3) where |£ | p = Sup x j | ^j(x) | p for the coefficient ^(x) in Xm-jYj of with x running over p"°°Z. Thus O^ is bounded and, by Proposition 3.2, we have a unique measure O^ extending the distribution £. Projecting down to the coefficient in ( • jXm~^

of O^, we get a measure (py. Now we want to show

dcp^j(x) = xMcp^o- To show this, we may assume that I h, | p < 1 by multiplying by a constant if necessary. We follow the argument given in [Ki] which originates with Manin [Mnl] and [Mn2]. For (|) G C(Zpx;A), take a locally constant function fa : (Z/p n ( k ) Z) x -^ A such that | fa-ty \ p < p" k and n(k) > k. Then we know that | O^((|)k)-O^((|)) | p < | O^ | pp"k < p"k and

122

4: Homological interpretation

z=l,(p,z)=l n k P

< >-i

I z=l,(p,z)=l P

j=0 z=l,(p,z)=l pn
= ap-n(k)

Wz)(X+zY)m^o(^y)

X

modpk

z=l,(p,z)=l

(

1 m

) "jYJ

mod p k ,

where ^j(x) is the coefficient of c(x)(£) in Xn"JYJ. Thus taking the limit making k -» °o, we see that J<j>dq>^j = J(|)(z)zJd(p^o(z) for all ty e C(Zpx;K).

(4)

Let N be a positive integer prime to p. We take % = 8 r i for each primitive character % modulo N ( 8 r i may not be p-integral but is bounded because (am+1-%"1(a))53C-i is p-integral as seen in §1). Then we write O^ as O x and compute for any primitive character $ of (Z/p r Z) x the integral ](|>dx. F ° r e a c n fa b\ triangular matrix a = 0 , , we let a act on C by a(z) = (az+b)/d, and for any differential form co on C, we write cc*co for the pullback of co by the action of a. We see that, if <>| * 1, then %$ is primitive and f

d O

V

x

r(V 0Yl V U

Z(p) r [ P 0

-

I AU

jjG(x)G«l))Jcof(rt)-i(X-zY)mdz

j=0 m

= -Z(p)rG(%)G((|))G«)Z)-1X N-jL(-j,x<|))(Ij1)xm-jYJ (see (2.2)) j=0 m j=0

v J 7

4.4. Another construction of p-adic Dirichlet L-functions

123

Here we have used the formula

(6)

Gfe)G(« = E u m o d N ,

vmodpI X(u)4Xv)e(£ + f)

= XumodN,

Now assume that <> j is trivial. Then we see that

This shows that

J

(7)

= - f N-J(l-X(p)pj)L(-j,x)(I°)xm-JYJ. X J

j=o

'

Now we assume that % - id and N = q * p is a prime. Then we take ^ = i(coo(f-f I q)) and write cpid forty^.Replacing c° by c° introduced in §2 to avoid singularity at 0, we see from (1.12) that (jfy is bounded and fe(l-^1(q)q-J-1)L(-j^)(I|1)xm-JYJ

r

(8)

-Ld*,,=i

J

if $ * id,

VJ / m X

[aX m +X J i : i (l-q-J- 1 )(l-x(p)p j )C(-j)(^)x m -JYJ if 0 = id,

for a suitable a e Q. Thus projecting down to the coefficient of ( i jXm in 3>x, we get by (4) a measure qfy satisfying, for all characters (j): (Z/prZ)x -> Kx, (9)

K)

-U(z)zidcp%(z) = / ^ N ) N ( l x 0 ( p ) p ) L ( j , ^ )

J9W

W;

id^-l^-l-JXl^)^)^^)

if X * id,

if

%

= id.

Here at first sight, the measure (p^ looks to depend on L(m;A) or m. However the evaluation formula (9) does not depend on m. Note that every function on f : (Z/p r Z) x -> K can be written by Lemma 2.3.1 as

f(g) = cpCpO'^xfWX^^"^) = ^(pO'^WgMZxfWcKx-1)}. Thus every locally constant function is a linear combination of finite order characters by (3.1). Since the space of locally constant functions is dense in the space of continuous functions (Proposition 3.2), each measure is determined by its value on finite order characters. This shows the independence of (px with respect to m. Summing up all these discussions, we get

124

4: Homological interpretation

Theorem 1. Let p be a prime and N be a positive integer prime to p. For each primitive Dirichlet character % ^ id modulo N, we have a unique p-adic measure | of Zpx and j e N, we

By the evaluation formula in the above theorem, we conclude that (pq = -£q-i on Z p x for the measure £a defined in Theorems 3.5.1 and 3.9.1. We now define the p-adic Dirichlet L-function for each primitive character % : (Z/NprZ)x —> Kx, writing %N (resp. %p) for the restriction of % to (Z/NZ)X (resp. (Z/prZ)x), by (s

Y)

(x)

=

H ^ ' J H Thus we get a generalization of Theorem 3.5.2:

if

%N

^ id,

if %N = id.

Theorem 2. For etfc/* primitive Dirichlet character % : (Z/Np r Z) x -» Kx %(-l) = 1, f/zere exwtt a p-adic analytic function Lp(s,%) on Z p «/ % ^ id on Zp-{ 1} if % = id ^MC/Z r/zar Lp(-m,%) = (l-%co-m-1(p)pm)L(-m,xco-m-1) /or all m e N.

Chapter 5. Elliptic modular forms and their L-functions A modular form of weight k with respect to SL2(Z) is a holomorphic function on the upper half complex plane # = {z G C | Im(z) > 0} satisfying the functional equation for all I* J e SL 2 (Z). Thus it is invariant under the translation f(z) = ^°°__ oo ane(nz)

ZHZ+1

for

and has Fourier expansion:

e(z) = exp(27tV^lz).

We always assume that an = 0 if n < 0. A typical example of such a modular form is given by absolutely convergent Eisenstein series E'k(z) = J / ( m n ) (mz+n)"k for every even integer k > 2, where Z' means that the summation is taken over all ordered pairs of integers (m,n) but (0,0). The Fourier expansion of E'k(z) is well known (we will verify the expansion later): ,, v _ , , ^2(27i>Ti)k\1T?l , , o.lrn ^k\z)

-

& k\z)

—(\r i \ t

~ *

SV^-K) + Z,

i^k-H 1 1 )^ »

where a m (n) = Z 0 < d | n d m is the sum of m-th powers of divisors of n. In this section, we study the complex analytic theory of modular forms. To each holomorphic modular form f = ^ anqn, we associate a Dirichlet series

and with each pair of modular forms f and g = ^°°_

bnqn, we also associate

another Dirichlet series

Then, we will study algebraicity properties of these modular L-functions later in this chapter and Chapters 6 and 10.

§5.1. Classical Eisenstein series of G L ( 2 ) / Q A subgroup of SL2(Z) is called a congruence subgroup if it contains all matrices a =

mod NM2(Z) in SL2(Z) for an integer N > 0. We study here

Eisenstein series for congruence subgroups of SL/2(Z). In this book, we are mainly concerned with modular forms with respect to the following type of congruence subgroups: for each positive integer N fa

b>1

SL 2 (Z) | c e N Z |

126

5: Elliptic modular forms and their L-functions I"i(N) = jl c

A e SL 2 (Z) | c e NZ and d = 1 mod Nk

In particular, we are interested in the case where N = p r for a rational prime p. We write A for one of these groups. A more detailed study of classical Eisenstein series for general congruence subgroups can be found in [M, Chap.7]. Each mafa b\ trix Y = I d I w i t n det(Y) > 0 acts on the upper half complex plane # = {z E C | Im(z) > 0} via the linear fractional transformation z h-> Y( Z ) = S^r- Then for any given function f on tt> we define an action of Y on f by f I kY(z) = det(Y)k-!f(Y(z))(cz+d)-k. For each positive integer k, the space of modular forms fA4(A) of weight k for A consists of holomorphic functions f on 9{ satisfying.the following conditions: (la) f | k y = f for all ye A; (lb) For each oce SL2(Z), flkOC has the following type of Fourier expansion of the following form: f | k&(z) = ^ n ^ Q a(n,f | ka)e(nz), where n runs over a fractional ideal aZ in Q. A modular form f is called a cusp form if a(0,f|ka)=0 for all a e SL2(Z). We write as 5k(A) the subspace of ^ ( A ) consisting of cusp forms. Let % : (Z/NZ) X -> C x be a character. Then we write ^k(Fo(N),%) (resp. 5k(r0(N),%)) for the subspace of MtfTiQ*)) (resp. 5k(Fi(N))) consisting of functions satisfying the following automorphic property: f k

l ( c d) = ^ ( d ) f

for

(c d ) G

r

°(N)'

Since the unipotent matrix I Q lj is contained in A, every modular form f in is invariant under the translation by 1; that is, we have f(z+l) = f(z). Since e(z) = e(w) if and only if z = w mod Z, we can consider f as a function of q = e(z). Thus the above condition (lb) means that f as a function of q is analytic at 0 and has the following Taylor expansion f(q) = Z°° a(n,f)qn. n=0

A typical example of a modular form of even weight k > 2 is given by the Eisenstein series which is defined by the absolutely convergent infinite series Ek(z) = I ' ( m , n ) (mz+n)-\ where "L"' indicates that the summation is taken over all ordered pairs of integers excluding (0,0). By definition, it is clear that E'k e iA4:(SL2(Z)) if the above summation is absolutely convergent, since , az+b (

+

5.1. Classical Eisenstein series of

GL(2)/Q

127

We consider slightly more general series for any non-trivial primitive Dirichlet character % : (Z/NZ) X -» C x : This series is non-trivial only when %(-l) = (-l) k because X^-nX-mNz-n)* = x(-l)(-l)k%-1(n)(mNz+n)-k. One can easily verify that this series is absolutely convergent if k > 2. We see that

= Z'(m,n) X'1(n)((mNa+cn,mNb+dn)[j])-k(cz+d)k. Rewriting (mNa+cn,mNb+dn) as (mN,n) for

fa b\ . e Fo(N) (because c is

divisible by N), we have ^ ; % ) = %(d)E'k(z;x)(cz+d)k. Thus E'k(z;%) satisfies the automorphic property defining the elements of ^4(Xo(pr)>%)- We now compute the Fourier expansion of E'k(z;%). We use the formulas (2.1.5-6): Z 1+

" Zr=i {(z+n)-1+(z-n)-1}=^cot(7Cz) = 7 i V : : l { - l - 2 X ^ 1 e(nz)}, where e(z) = exp(27C V-Iz). This series converges absolutely and locally uniformly with respect to z, and hence we can differentiate this series term by term k-times by (27ci)~1 -^, and we get

Then we have N

Xx" 1 (r)XLi Xr = -~ (mNz+r+nN)-k r=l <*> N m=l j=l

= 2L(k)X-1) + 2N- k ( -f^ ) k £ £ x -»0)i n"e(n(mz4 ^ - ^

m=l

jj=l l

n=ll

128

5: Elliptic modular forms and their L-functions

We now compute Z]^1%"1G)e(fj)- If

n

is prime to N, then by making the

variable change nj \-> j in the summation, N

.

where we write G(%~1) =

N

S^X^OM-NT)-

(If X is trivial, then 1+G(id) is the

sum of all N-th roots of unity and hence 0. Thus G(id) = -1.) If % is primitive, then as seen in Exercise 2.3.5 and (4.2.5a,b), we have Anyway G(%) ^ 0. If n is not prime to N, then we have Sj=1%"1(j)e(j4) = 0 by Lemma 2.3.2, because % is primitive. Thus we know, for primitive %, that E' k (z;x) = 2L(k,x" 1)

l

^

'-

[m=0n=0,(n,N)=l

By the functional equation (Theorem 2.3.2), we know that

1

1

1 We put Ek(z;x)={2N-kG(x- )^^}" )^^} 1E' E'kk(z;x) (z; and

where we agree to put %(d) = 0 if d has some non-trivial common factor with N. Then we have oo

Proposition 1. We have E k (z;x) = 2' 1L(l-k,%) + ^

ak-i,x(n)e(nz) if % is

n=l

primitive modulo N or % = id. We note here that Ek(z)-pk"1Ejc(pz) has the following Fourier expansion:

p

n=l

where i p is the identity character modulo p (therefore, ip(pn) = 0 for all n and i p (n) = 1 if n is prime to p). Although we have only proved this proposition under the assumption that k > 2, the assertion for non-trivial % is true even for k = 1 and 2. We will see this in Chapter 9.

5.1. Classical Eisenstein series of GL(2)/Q

129

We now want to compute a slightly different series for a Dirichlet character % modulo N (we allow here % to be imprimitive): oo

oo

G'k(z;%) = £(».„) X(m)(mz+n)-k = 2 £ x(m) X (mz+n)"k = 2 £ X(m) £ ^ ^ 1 m=l

n=—oo

nk-1e(mnz) = 2 ^ ^ f ;

n=l

a'k-i.x(n)qn.

n=l

where we write a'm>x(n) = Xo
n=l

For x =

o -

, we see easily that

E'k(z;%) | kT = N-1z-kX(m,n) X ^ W d n N ^ f n ) " k = N ^ G U z ; ^ 1 ) . We in particular have, for a primitive character %, Ek(z;%) | kx = %(-l)Nk-2G(x)Gk(z;%-1), where we have used the fact that G(%)G(%'1) = %(-l)N. We note this fact as Proposition 2 (Hecke). Let % be a Dirichlet character modulo N with k Z (-1) = (-l) . We have Ek(z;%) = 2-^(1-^%) + Y^=l

^k-i,x(n)qn for primitive %,

Gk(z;%) = 5k,2(27cy)-1+5k,i2-1L(0,%) + £ " = 1 a' k .i, x (n)q n for any %, 8 k j = 1 or 0 according as k = j or nof. Unless k = 2 and % zs1 r/ze identity character, the functions Ek(z,%) and Gk(z;%) ar^ elements in ^k(r o (N),x). Moreover we have Ek(z;%) | kT = %(-l)Nk-2G(%)Gk(z;%-1). Proof. We only need to prove that E'k(z,%) and G'k(z,%) satisfy (lb). We prove this assuming k > 2 and complete the proof in Chapter 9 (see Theorem 9.1.1 and (9.2.4a,b)) dealing with the case where k = 1 and 2. We consider a slightly more general series: for a pair (a,b) of integers E f kfN (z;a,b) = Then it is easy to see that E'k>N(z;a,b) | koc = E' k> N(z;(a,b)a) and

E'k(z;z) =

E

1

130

5: Elliptic modular forms and their L-functions

G'k(z;%) = E X Thus we only need to prove that E'k,N(z;a,b) has no negative terms in its Fourier expansion. We see by definition that E'k,N(z;a,b) is equal to

5a

In*

n=b mod N

^

^ , /mz+b

V / i \k

msamodN,m>0n=-» \

/

xf

XT

msamodN,m
where 6 a = I 1 i f a ^ 0 mod N, [0 otherwise. By (2), we have for a constant c ^ 0 °° (\m I

Z

n=-oo V

Yk

z±b N

hn

= c J

v^ ^n

k

i

/ e(n(

Am z ± b x )} N '

n=l

This shows that the Fourier expansion of E'k,N(z;a,b) does not have terms in e(nz) with n < 0. As a byproduct of the above calculation, we get (3)

The constant term of the Fourier expansion of Gk(z;%) | kOC vanishes for every a =

e Fo(p), if % is primitive modulo p r and k > 2.

We prove the following facts for our later use (4a) / / k > 2, then I <^k-i,%(n) I ^ ^ ( k - l ) ^ " 1 and I o'k-i,x(n) | ^ ^ ( k - l ) ^ " 1 . (4b) If k = 1 or 2, for any e > 0, there is a positive constant C such that I ak_i,x(n) | < Cnk"1+E and \ a' k .i, x (n) | < Cnk"1+e. We compute where n = n p | n p e(p) - Then we see that

-p 1 - k )SC(t-l)n k - 1 if k > 2 . Since I o\.\y%{n) \ < \ a'k-i,id(n) I = I C7k-i,id(n) I, the assertion is also true for a'. The above argument yields

5.2. Rationality of modular forms
131 if

s

> 2'

Since I ai,id(n) | < Iai + e , i d (n)| < C(l+e)n £ for all e > 0. This proves (4b) for k = 2. When k = 1, we have

I aOiid(n) I = I I p | n(e(p)+D if n = I I p | n p e(p) . I ao,id(n)n-£ | < I I p | n(l+e(p))/pe(P)£.

Thus

Since p e ( p ) £ > 1, we see, if n > 2, that e(p) .+e(p)) 1 p e(p)e

If p > 2

1/£

~

p e(p)e

£

, then p > 2 and In

_^

pe(p)E

"

2e(p)

Combining these two inequalities, we see that I G0,id(n)n-e | <

1/£ exp(2 1/£ /elog2) < Cn £ .

§5.2. Rationality of modular forms We first deal with the rational structure of the space #4(SL2(Z)). We introduce several modular forms with integral Fourier coefficients to study the rational structure in an elementary manner. For k = 4, 6, 8, 10 and 14, we put G k (z) = 2C(l-2k)" 1 E k = 1 + C k ^ i

a k .!(n)q n e fWk(SL2(Z)).

By the actual values of £(l-2k) given after Theorem 2.2.1, we have k

ck

4 240

6 -504

8 480

10 -264

14 -24

We could have defined G12 in a similar manner, but then it would have had the prime 691 as the denominator of its Fourier coefficients. We further put G 0 (z) = 1 and A(z) = g|(z) - 27 g 3(z) for g2 = G 4 /12 and g3 = G 6 /216. The function A is called the Ramanujan's A-function (or the discriminant function). Computing the constant term of A, we see that A e 5i2(SL2(Z)). It is known that A does not vanish on !H and even has the product expansion

A(z) = q f J Q ^-q")24

132

Elliptic modular forms and their L-functions: Chapter 5

Let X = (SL 2 (Z)\#)U{~} (see [Sh, 1.4, 6.1] and [M, §4.1] for how to give a / standard structure of Riemann surface on this space). Put J = G4/A. Then J has the q-expansion of the form: q"1 + X°° c n q n with c n e Z. The function J n=0

gives an identification of X with the projective J-line P^J) [Sh, Chapter 4]. We define for each positive even integer k =

r =

J[k/12], if k s 2 mod 12, l[k/12] + l, otherwise,

where [q] for a rational number q denotes the largest integer not exceeding q. Put s(k) = k-12(r(k)-l). Lemma 1. The equation 4a+6b = k-12(r(k)-l) = s(k) has one and only one non-negative integer solution for each even integer k. Proof. If k = 2 mod 12, then k-12(r(k)-l) = 14; in this case, the unique solution is (a,b) = (2,1), and if k 4 2 mod 12, then k-12(r(k)-l) < 12 and the uniqueness and the existence of the solution can be checked easily. We list all the solutions in the following table: k mod 12 S

a b

0 0 0 0

2 14 2 1

4 4 1 0

6 6 0 1

8 8 2 0

10 10 1 1

We choose the unique solution (a,b) as above for a given k and define ^ = (G 4 ) a (G 6 ) b+2(r ' 1 ' i) A i e af k (SL 2 (Z)) for i = 0, 1, —, r-1 (r = r(k)). Note that hi e 3Vfk(SL2(Z)), and hi has the q-expansion (q = e(z)) with coefficients in Z of the following form: hi = q1 + £ ~ = i + 1 b nq n w i t h b n ^ Z. This shows that hi are linearly independent. For any subring A of C and each congruence subgroup F, we put ) = {fe Mk(T) I a(n,f) e A for all n}, Sk(T;A) = Theorem 1. We have dim c (^ k (SL 2 (Z)) = rankz(f7Hfk(SL2(Z);Z)) = r(k) and for any subring A of C, f*4(SL2(Z);A) = ^k(SL2(Z);Z)zA, 5k(SL2(Z);A) = 5k(SL2(Z);Z)zA. Moreover {ho, •••, h r .i) (resp. {hi, •••, hr_i}j form a basis of f^"k(SL2(Z);Z) (resp. 5k(SL2(Z);Z)j over Z. 1

X

I am indebted to Y. Maeda for the construction of the basis (hi).

5.2. Rationality of modular forms

133

Proof. If we know that the dimension of the space of modular forms is less than or equal to r(k), then hi gives a basis of fA4(SL2(Z);A) over A and everything will be proven. Let us show the inequality dim c (44(SL 2 (Z))) < r(k). Put s(k) = k-12(r(k)-l). Recall that k = 0,2,4,6,8,10 mod 12. We put

s(k) = 0,14,4,6,8,10

according as

T k (z) = G14-s(k)(z)A(z)-r. Then T k is holomorphic everywhere on !H and we have the following q-expansion of Tk (q = e(z)): (1)

T k (z) = ck,rq~r+ ••• +c k i o+--- with ck)jG Z (ck,r = 1).

Since the weight of T k is given by 14-s(k)-12r(k) = 2-k, for each f e fA4(SL2(Z)), fTk(z) is of weight 2. Thus the differential form CO = fTk(z)dz = —?=fTkdq/q satisfies y*co = co for all y e SL2(Z) and is holomorphic on fH. That is, co has a singularity only at infinity. By construction, the singularity of co at infinity is a pole of order at most r+1. Let C0m = JmdJ which has a pole of order m+2 at infinity and which is holomorphic outside infinity. Let 8 be a meromorphic differential form on X whose singularity at infinity is a pole of order n and is holomorphic outside infinity. Then n > 2 since deg(8) = 2g-2 = -2 for the genus g = 0 of X. In fact, this can be proven as follows. Since X is a Riemann sphere with coordinate J, at any point x e X , J-J(x) is a local parameter at x. Therefore dJ has order 0 at x. On the other hand, at infinity, dJ has order -2 and hence deg(dJ) = -2. For any morphism f : X —» PX(C) of algebraic varieties (or Riemann surfaces), the numbers of points in the fiber at 0 and oo are equal (counting with multiplicity). Thus deg(f) = #(points over 0) - #(points over °o) = 0 and therefore, writing 8 = fdJ for a suitable function f: X —> P ! (C), we see that deg(8) = deg(fdJ) = -2. Thus, subtracting suitable constant multiples of the coi's from 8, we may assume that 8-(boCOo+-*-+bn-2COn-2) has at most a simple pole at infinity and is everywhere holomorphic outside infinity; that is, deg(8 - bocoo + ••• +bn_2C0n_2) £ -1. This implies 8 = boCOo+---+bn_2COn_2. Applying this argument to co, we know that co can be written as boCOo+-**+br_iCOr_i. Via the map f H» co = fTkdz, we can embed f*4(SL2(Z)) into the space of differential forms generated by coo, • • •, GVi and thus we get dim c (*4(SL 2 (Z)))
134

Elliptic modular forms and their L-functions: Chapter 5

Since dimc(#4:(SL2(Z))) = r(k), the first r(k)+l q-expansion coefficients of any f e f7Vfk(SL2(Z)) have to satisfy a non-trivial linear relation. We can make explicit this linear form: Corollary 1 (Siegel [Si]). / / f = X~ =o a n q n €= *4(SL 2 (Z)), then Ck.oao + "• + Ck,rar = 0 (r = r(k)) and where Ckj are the Fourier coefficients of Tk in (1).

Ck,o ^ 0,

Proof. Note that com = J m dJ = -L-HL—dq and

ai

m +1

jf

dq

-r uq

m

1

= 27u'mq ), and hence the coefficient in q" of com is always 0. The

constant term of Tkfdq is given by Ck,oao + •** +«k,rar, which shows +—•- Ck,rar = 0. Now we prove that Ck,o ^ 0. Note that

fl

n

n=l

n=l

^=0

Therefore the coefficients of A"r are all positive. On the other hand, the coefficients of Go, G8, G4 are positive and hence Ck,o ^ 0 when k = 2 mod 4. Now suppose that k = 0 mod 4. Then s(k) = k-12(r(k)-l) = k m o d 4 and we can write s(k) = 4t with t e Z. Now consider co = (27tO"1Gi4A"1dq/q = GuA^dz which has a pole of order 2 at infinity and is holomorphic outside infinity. Therefore we can write co = cdJ with c * 0 and thus G14A"1 = C-T- or Gu = cA^

because r(14) = 1. This shows that Tk = cGi 4 . s ( k )A 1 " r (Gi4)' 1 ^ .

Since fAfi4(SL2(Z)) is of dimension 1, we know that Gi4_s(k)Gs(k) = G14 and GS(k) = (G4)1, comparing the constant terms. Then, we know that

By replacing G4 by (AJ)1/3 in this formula (G43 = AJ by the definition of J), we have X _ c^l-r-t/3j-t/3dJ

=

3C ^l-r-t/sdJ1"

Note that J H t / 3 ) = (G4)3-tA(t/3)"1. Then we have

dz

~A

dz

5.2. Rationality of modular forms

135

d(G4)3-lA-r _ d(G4)3-tA(t/3)-1A1-r-t/3 "" A

dz

+ (CJ4)

A

dz

and dz

"

r

Therefore, we see that T Tk =

3c d , r 3-t A .rx ^ (3r+t-3)cr3-tdA-r 3^dz"(G4 A ) + (t-3)r G 4 "dT •

Since ^-(G4"tA"r) has no constant term, we look at the second term of the above formula. Since the coefficient of qJ in —T—- for j < 0 is negative, and the coefficients of G4~l are all positive, we know that the constant term of the second term is negative. This shows that ck)o < 0. Applying the linear form in Corollary 1 to Ek(z), we have Corollary 2.

r ^ l - k ) = -Ck^X]!?

a

k-i(n)c k J for all 2 < k e 2Z.

Let me briefly explain how Siegel applied Corollary 1 to show the rationality of the values of the Dedekind zeta function of a totally real field F of degree d. Let a be an ideal of the integer ring O of F and consider the following Eisenstein series for z e M (see Theorem 2.7.3): for even integer k. Here N(mz+n) is the product Tlairxfz+n0) taken over all embeddings a of F into R, and (m,n) runs over all equivalence classes of ordered pairs of numbers (m,n) in a under the relation (m,n) - (m',n') if m' = em and n' = en for e G Ox. This series is absolutely convergent if k > 2. We can verify that Ek(z;a) is a modular form of weight k[F:Q] with respect to SL2(Z), and we can compute its Fourier expansion (see Theorem 9.1.1). Even if one replaces a by Xa for X G F x , each term of Ek(z;a) NF/Q(a)kN(mz+nyk = N F / Q (M k A^mz+?tn)- k does not change, and thus, Ek(z;a) depends only on the ideal class of a. To simplify a little, we define

Ek(z) = £ a E k (z;a), where a runs over a set of representatives of ideal classes of F. Then we have

136

Elliptic modular forms and their L-functions: Chapter 5

n=l

where i3- is the different of F, 2; runs over all totally positive elements in ft"1, D = A^F/Q(I^) is the discriminant of F, £F(S) = Z ^ F / Q ( ^ ) " S is the Dedekind zeta function of F and Ok-i(£) = Z Corollary 3 (Siegel). For each even positive integer k, we ~n = ~

c

kd,0

G

^

This fact also follows from Corollary 2.7.1 proved by Shintani's method by the functional equation of £F(s) (Corollary 8.6.1). Now let us note another application of Corollary 1. Let p be an odd prime and consider the Eisenstein series f = Ek(z;coa) e f7t4(ro(p),coa). Let b be the order of coa. Then fb is an element of #4b(ro(p)). We choose a complete representative set R for Fo(p)\SL2(Z) and define Tr(g) = EyeRg I kY for g e fA4(Fo(p)). Then obviously Tr(g) e 3tf"k(SL2(Z)). We now want to compute Tr(fb). Lemma 2. We can take as R the set of the following matrices: the identity ma(1 ft (0 -I) trix I2 and 8j = 8 L A for 8 = L Q with j = 1, ••• , p. // g(z) = E°° Aane(nz) an^ g | k 8 = n=u

Tr(g) = Proof. Take y = I

J e SL2(Z). If c is divisible by p, then y is in Fo(p)

and in the left coset of I2. Thus we may assume that c is prime to p. Take an integer j in the interval [l,p] such that cj = d mod p and consider The entry at the lower left corner of this matrix is cj-dwhich is divisible by p by definition of j . Thus y e Fo(p)8j. If y and y' = I , hence

d,

I are in the same left coset of Fo(p), then cd' = c'd mod p and

SL 2 (Z) = UjFo(p)8jUFo(p)

is a disjoint union.

Note that

I j U g I k 5j = i ; = 0 b n l P = 1 e ( ^ ) = pb 0 + S; =1 pb np e(nz). From this, the formula in the lemma is obvious.

5.2. Rationality of modular forms Recall that for the Eisenstein series n

137

Gk(z;%) = £°° G'k_i,%(n)e(nz)

w

^

m

tf'm,x( ) = £o
(o -n where x = J put

0

and % = coa for the Teichmiiller character co. Note that

and therefore

f = Ek(z;%)

and

Ek(z;x) I k 8 = (-l)V 1 G(%)G k (|;x" 1 ). Thus if we

g = ( - l ) k p - 1 G ( x ) G k ( - ; x - 1 ) , then fb | k 5 = gb = P

£°° bne(—-) where b n for all positive n are algebraic numbers. Write f as X + F(q) b

with

b

X = 2- 1 L(l-k,co a )

and

F(q) e qQ(co a )[[q]].

Then

n

f = X + S°° an(X)q , where an(X) is a polynomial of X of degree strictly n=l

less than b with coefficients in Q(coa). Thus by Siegel's theorem, we have an equation with coefficients in Q(coa):

c kb ,oX b + X j b ) ckb>j(aj(X)+pbjp) = 0, where the degree of aj in X is strictly less than b. Thus the above equation is non-trivial, and we get another proof (valid only when k > 1) of the following fact: Proposition 1. 2~ 1 L(l-k,co a ) for coa(-l) = (-l) k .

k > 0 is an algebraic

number if

We now want to show that 2"1L(l-k,coa) e Q(coa). For that purpose, we introduce the transformation equations. For each modular form f in Mk(T), we fix a representative set R for F\SL2(Z) and define P(f;X) = n T e R ( X - f I kY) = X<1 + s i ( f ) x d " 1 + - + sd(f)Note that Sj(f) e 5tfkj(SL2(Z)) and P(f;f| k y) = O for any y e SL 2 (Z). For each f E M^(T)y we formally define the conjugate f° = E°° a(n,f) a q n as an n=0

element of C[[q]] for any automorphism a of C. Since the map defines a ring automorphism of C[[q]], if we put = X d +si(f)°X d - 1 +-"+s d (f) a , then Pa(f;f°) = O in C[[q]]. Proposition 2. For f e !Wk(SL2(Z)), we have f° e ^ k (SL 2 (Z)) for each a e Aut(C). In particular, Sj(g)a e ^ k j(SL 2 (Z)) for any ge

138

Elliptic modular forms and their L-functions: Chapter 5

Proof. Write f = coho+---+cr_ihr_i for the basis hj as in Theorem 1. Then fa = c 0 0 ho+-+c r .i°h r .i€ *4(SL 2 (Z)) because hj e Z[[q]]. Let fM"(C) = ©"T fAfk(SL2(Z))

and for any subring A of C, we put

= fW(C)DA[[q]]. Note that 0f(C) is a graded algebra and f = ®kfk H> f° = 0kfk a for a e Aut(A) is a ring automorphism of Let A(A) be the quotient field of Proposition 3. If f e fMk(F) for a subgroup T of finite index of SL2(Z), then for any a e Aut(C), f° e ^k(A) for a normal subgroup A offinite index in SL2(Z). Proof. Let F (resp. F°) be the set of all roots of P(f;X) (resp. Pa(f;X)). Each root g of Pa(f;X) is not only a formal Fourier series but it converges on m because it is a root of a polynomial with function coefficients. Thus for y G SL2(Z), we can consider g | kY, which is a root of P a (f;X) | y = X d + si(f)° I kYX*"1* ... + s d (f) a | kd y = 0. because The left-hand side of this equation is equal to P a (f;X) a a Sj(f) e fA4j(SL2(Z)). Thus F is stable under SL2(Z). Let A be the subgroup of SL2(Z) which fixes all elements of F a . Then the action of SL2(Z) on F° gives a representation of SL2(Z)/A into the group of permutations of elements of F a , which is a finite group. Thus A is a normal subgroup of finite index in SL 2 (Z). The following fact is clear from Corollary 2.3.2, but we shall give another proof of the fact: Theorem 2. The value L(l-k,%) for %(-l) = (-l) k is an element of Q(%), and for each ae Aut(C), L(l-k,%) a = L(l-k,% a ). Proof. Let a be an element of Aut(C). Then for f = Ek(z;%), all Fourier coefficients of f° but its constant term coincide with those of g = Ek(z;%a). Thus C = f°-g is a constant, which is a modular form of weight k for a subgroup A of finite index in SL2(Z) (if f° is modular with respect to O and g with respect to F, then O/Ofir = OF/F, which is a finite group). Therefore for some y =

e A with c * 0 , CI kY = (cz+d)"kC = C. Since cz+d* 1, we

know that C = 0. Thus (L(l-k,%))a = L(l-k,x a ). This shows the assertion. The above proof of the rationality of the Dirichlet L-values using the action of a e Aut(C) is due to Shimura.

53. Hecke operators

139

§5.3. Hecke operators In this section, we shall introduce the Hecke operator T(q) as an endomorphism

n o\

of ^k(Fo(pr)>%)- We consider the double coset F L F for primes q with F = Fo(N) for a prime power N = p r . Then we can decompose, for f l 0\ a = I0 I, F a F = UiFai as a disjoint union; actually, an explicit left coset decomposition is given by ;

;)(;;)

to

N.

FaF =

IE. r l» pJ A proof of the above decomposition is given as follows. 7 = 1

d

ible by y = r by

Take any

e M2(Z) with ad-bc = q. If c is divisible by q, then ad is divisq,

q

and thus one of H

q but

J e SL2(Z) L

a is prime to

ua = b mod q, YI

a

I

e

and

d is divisible by

q.

We have

J if a is divisible by q. If d is divisible

q, then by choosing

u e [l,q]

such that

SL2(Z). If c is not divisible by q but a is divis-

(0 -I) ible by q, we can interchange a and c by multiplication by 8 = L Q I on the left. If both a and c are not divisible by q, then by choosing u so that

f

ua = -c mod q, we know that the lower left corner of I and is divisible by q. This shows (la) for

1

Jy is equal to ua+c

F = SL2(Z). The case of

r

F = Fo(p ) can be similarly treated (see [M, Lemma 4.5.6]). Note that for any element I

d

in Fo(pr)^Fo(pr)J c is divisible by p r and a is prime to p

(because every element of Fo(pr) is upper triangular mod pr). Exercise 1. Give a detailed proof of (1) when F = Fo(pr)We define the Hecke operator T(q) on 3Wk(Fo(N),%), using the disjoint decompositions F a F = UiFai and F a T = UiFpi (a 1 = det(a)a" 1 ), by (lb)

f | T(q) = Ii%(ai)f | k ai and f | T*(q) = liXi^f

I kPi,

140

5: Elliptic modular forms and their L-functions

(fa bY) where jn = %(a) (a is always prime to N by (la)). Here we understand that % is trivial when F = SL 2 (Z). Then obviously T(q) and T*(q) gives an operator acting on fA*k(F) if F = SL2(Z). If ye Fo(N), then = IliFai = IliFaiY and thus f I T(q) | k y = Xi%( a i) f I kOCiT = X(y)"1Xi5C((XiY)f I kOCiY = X(yYlf I T(q). -1

Here note that

%(y)

b\

fa

i

= %(a)

= %(d) for

I

d

I e Fo(N)

because

a d s 1 modN. This shows that f I T(q) e fMk(F0(N),%). Similarly, we can show that T*(q) preserve #4(Fo(N),%). Now we want to compute the Fourier expansion of f|T(q). When f e fl4(Fo(pr)>%) for r > 0 , then

f |T(p) = p" x X f ( ^ ) = P - 1 E a(n,f)e(^)|; e(y) = £ a(np,f)e(nz). u=l

n=0

u=i

n=0

As for T(q) for q * p, we have

( As the contribution of the term J J ^ Fl Q

I to the coefficient of e(nz), by the

fq 0) same computation as above, we get a(nq,f). For the remaining term Fl 0 j I, we get %(q)qk-h(n/q,f). (lc)

Thus we have for f e fMk(ro(N),%) (N = p r ), a(n,f | T(q)) = a(nq,f) + xCqJq^adi/q.f),

where we have implicitly assumed that a(n,f) = 0 if n is not an integer and %(q) = 0 if q IN. If q and r are two distinct primes, then a(m,f | T(q)T(r)) = a(mr,f | T(q)) + x f r ^ a W I T(q)) = a(mrq,f) + x(q)qk-1a(mr/q,f) + xWi^^

^ V

which is symmetric with respect to r and q and hence (2)

T(r)T(q) = T(q)T(r) if r and q are different primes.

By the above formula, we have (3)

a(m,f | T(q) 2 ) = a(mq2,f) + 2x(q)qk"1a(m,f) + %(q)2q2(k"

We now define the operator T(qe) for e > 1 inductively by

(4a)

W+1)-{

T(q)6+1

i f 6 1

ilN,

[TCqFCq^-xC^qk^TCq - ) otherwise,

5.3. Hecke operators

141

where we define T(l) to be the identity map. Then we see from (2) that T(qe)T(/s) = T(/s)T(qe) for different primes / and q. More generally, we can define T(n) for each positive integer n by

(4b)

T(n) = Ylji(qeiq))

if n = I ^ q ^

for

Primes ^

Then we can write down the Fourier expansion of f | T(n) explicitly (see [Sh, (3.5.12)]) as

(5)

a ( m , f | T ( n ) ) = £ o < b | (m>n) ^

V

M

Define a semi-group A (depending on N) by (6a)

A = {a e M 2 (Z) I det(a) > 0} if T = SL 2 (Z) and

A = {a = L

J e M 2 (Z) | p|a, p r | c and ad-bc> 0} if T = r o (p r )-

Let R be a complete representative set for I\{ a e A | det(a) = n}. Then it is known that in fact which is the original definition of Hecke (see [Sh, III] and [M, IV]). In particular, on 5k(ro(pr)>%) (r > 0), we can easily check that

(6b)

f|T(p s )= X f | k P U=l

^

Let A be a subalgebra of C. By (5), fA4(I\%;A) and 5k(F,%;A) are stable under the Hecke operators T(n) if A contains Z[%]. Here we write ^k(r,x;A) = fMk(r,%)nA[[q]] and 5 k (I\x;A) = 5 k (r,%)nA[[q]]. Let V be an A-submodule of any one of the above spaces stable under T(n) for all positive integers n. Then we define the Hecke algebra d(V) by the A-subalgebra of EndA(V) generated by T(n) for all positive n. Note that T(l) gives the identity on V and hence /t(V) is a commutative algebra with identity. Hereafter we suppose that A contains Z[%] if we consider f7Vfk(ro(N),%;A) and 4(TO(N)JC;A). We define H k (r 0 (N),x;A) = fi(fWk(ro(N),z;A)) and h k (r 0 (N),%;A) = Lemma 1. The space f7tfk(ro(N),%;C) is offinite dimension over C. Proof. By replacing T = Fo(N) by the kernel of %, we may assume that % is trivial. We fix a representative set R for r\SL2(Z) and consider the A-linear map M IM • MkfX) -> C * for each positive integer M given by IMCO = (a(n,f I ky))n,y, where (n,y) runs over all possible pairs with 0 < n < M and y e R for

142

5: Elliptic modular forms and their L-functions

which we have some modular form f with a(n,f I kY) ^ 0 and M* is the number of such pairs (n,y). The number M* is smaller than NM[SL2(Z):F] because we always have n e N -1 Z for the pairs (n,y) as above. Take M>r(k)+1 and suppose that IM (f) = 0. Let si(f) be the coefficient of X d-1 in P(f;X) = II Y eR(X-f IkY) for d = #R. Then we have a(n,Si(f)) = 0 if n < r(ki) < i(r(k)+l). Hence by the proof of Theorem 2.1, si(f) = 0 for i > 0. Thus P(f;X) = X d . Since 0 = P(f;f) = f1, we know that f = 0. In particular, we have < (r(k)+l)N[SL 2 (Z):F]. We define for the quotient field K of A "%(r o (N),%;A)={fe ^ k (F 0 (N),x;K) | a(n,f) e A if n > 0 } . Clearly we know that /%(Fo(N),x;A) z> f7tfk(ro(N),x;A), but they may not be equal. Theorem 1 (duality). Suppose that A contains Z[%]. Define a pairing

( ,) : Hk(r0(N),x;A) x mk(r0(N),x;A) -> A by (h,f) = a(l,f | h) e A. Then this pairing is perfect on /rck(Fo(N),x;A) and 5 k (Fo(N),x;A); in other words, we have isomorphisms p HomA(Hk(r0(N),x;A),A) = mk(r0(N),%;A), HomA(/%(Fo(N),x;A),A) = Hk(F0(N),x;A), Hom A (h k (r 0 (N),x;A),A) = 5k(F0(N),x;A), HomA(5k(r0(N),x;A),A) s hk(F0(N),x;A). Proof. Here we prove this theorem under the assumption that either N = 1 or A = C and later return to this problem for general N = p r and A. First assume that A = C. Write F = r o ( N ) . Since H k ( I \ x ; C ) is a subspace of Endc(flik(r,x;C)) which is of finite dimension, Hk(F,x;C) is of finite dimension. Thus we only need to prove the non-degeneracy of the pairing. Since a(m,f | T(n)) = I b | ( m , n ) X(b)bk"1a(mn/b2,f), we see that = 0. Thus f I h = 0 for all f and hence h = 0 as an operator. This proves the assertion for A = C. The above argument is valid as long as A is a field and both Hk(F,x;A) and mk(F,x;A) are of finite dimension over A. In particular, the theorem is true for A = Q when F = SL2(Z) by Theorem 2.1. Now we want to show the theorem for A = Z and F = SL/2(Z). Since A is a principal ideal domain, it is sufficient to prove one of the following two assertions:

5.3. Hecke operators Hom A (H k (r;Z),Z) s mk(F;Z),

143

Hom A («k(r;Z),Z) = H k (r ; Z).

We shall show that Hom A (H k (r;Z),Z) = /%(F;Z). Since mk(F;Z)
H k (F;Z)® z Q = H k (r;Q).

The natural map from mk(F;A) into HomA(Hk(F;A),A) is injective, because if (h,f) = 0 for all h, then f = 0 as shown already. If 9 : Hk(F;A) -> Z is a linear form, we can extend (p to a linear form on Hk(F;Q) with values in Q by linearity. Then we can find f e /%(F;Q) such that (h,f) = (p(h) for all h. Then a(n,f) = cp(T(n)) e Z and hence f e 7%(F;Z). This proves the surjectivity. As for general A, since /%(F;A) = /%(F;Z)<8)ZA, by definition, we know that Hk(F;A) = Hk(F;Z)zA. Thus Hom A (H k (F;A),A) = Hom z (H k (F;Z),Z)®A = Hom A (m k (F;A),A) 2 Hom z (m k (F;Z),Z)®A 2 Hk(F;Z)(g>A 2 Hk(F;A), which finishes the proof. We consider the hermitian product (called Peters son inner product) on 5k(To(N)jc) defined by (7)

(f,g) N =

where O is the fundamental domain of Fo(N). First of all, we know that

f(y(z)) = f ( z ) ( c z + d ) k for y = ^

J j e F . Thus gf(y(z)) = gf(z)Icz+d1 2 k .

On the other hand, dxAdy = (-2i)"1dzAdz and thus y*(dxAdy) = I cz+d I "4(dxAdy). By taking the determinant of the formula bYz z^| _ ry(z) y(z)Ycz + d 0 ^

dj[l lj " t 1

1 j{ 0

cz + dj'

2

we know that Im(y(z)) = det(Y)|cz+d|" Im(z). Thus fgyk~2dxAdy for y = Im(z) is invariant under the action of F, and hence at least formally the above integral is well defined. By the same formula, | f(z)yk/21 for f e 5k(Fo(N),%) is a continuous function on Y = Fo(N)W. Since as a function of q = e(z), f vanishes at q = 0, on a small relatively compact neighborhood V of 0, f(q) = qF(q) with a holomorphic function F on V. Thus on the closure of V, (8a) I f(z)y^ 2 I is bounded on M and | f(z) I < O(exp(-2?cy)) as y -> 00. Writing the bound of I f(z)Im(z)k/21 as Co, we see that

144

5: Elliptic modular forms and their L-functions

| a(n,f) I = I f f(x+/y)e(-n(x+zy))dx | < C By taking the minimum of the function coe27111^"^2, we conclude that (8b)

I a(n,f) I < Cn k/2 for f e 5k(F,%),

where C is a constant independent of n. Thus by (8a) a* fgyk~2dx A dy for a G SL2(Z) is exponentially decreasing as Im(z) -» oo, and hence the integral (7) converges. Thus (, )N is a well defined positive definite hermitian product on 5k(Fo(N),%). More generally, for any subgroup F of finite index in SL2(Z), we can define (f,g)r for f , g e 5k(F) by the same formula (7) replacing Fo(N) by T. Note that if F 3 F and if [r:F] is finite, then for co = fgy k " 2 dx A dy with f,ge 5k(F), y*co = co for all ye F. On the other hand, if F = Uf% then F = UjYj^r1, and XIX = Llj Yj
(9) (f,g)r = EiJd,^" 1 )* 00 = Jo® = ( r : r ')( f 'S)r

for

r

For any a e A, the group F' = FPloc^Fa is of finite index in F. Thus for f5g e ^k(F), we have flkOce 5k(F') and g(fla)y k ~ 2 dxAdy =

^det(a)k~lf(a(z)))(a9zykyk~2dxAdy

= a*(gla l )f(z)y k " 2 dxAdy, where we have written j(

, ,z) = (cz+d) and a = det(a)a . Then we

have, for the fundamental domain *¥ of F , (f | a , g ) r = JyCt *(glal)f(z)yk-2dxdy = Ja(xF)(glal)f(z)yk"2dxdy = (f,g | because aQ¥) is a fundamental domain of a F ' = aF'a" 1 . Since (F:F')vol(O) = vol(Y) = vol(aOF)) = ^ 2 1 1 under the invariant measure y" dxdy, (FiF ) = (FtaF'a" ). We now want to compute (f I [FaF],g) for F = Fo(pr)- Decompose F a F = IIjFaj and write O for the fundamental domain of F in H Then we have (f | [FaF],g) = XjX(ocj)(f I a jf g) = Lemma 2. We suppose that #(F\FaF) = #(F\Fa l F) for F = F 0 (p r ). Then we may choose aj so that F a F = LIjFaj and F a l F = IIjFaj 1 .

5.3. Hecke operators

145

Proof. Since the involution i brings a right coset onto a left coset, by assumption, the numbers of the right cosets and the left cosets in FocF are the same. If FocF contains F£, and r\T, then ^ = Srjy for 8 and y in F. Then for C = S' 1 ^ = riy, TC, = FS ^ = F^ and £F = r\yT = T]F. Thus we can choose (Xi so that F a F = UiFoci = UiOCiF. Since i is an involution, this means that (FocF)1 = U Note that A = {a = ( a " l e M 2 (Z) Then for a = I

a

p|a, p r | c and ad-bc>0}.

, I e A, ad= det(oc) mod p. Thus if det(oc) is prime to

p, then %(al) = %(d) = X"!(a)%(det(a)) = %"1(a)%(det(a)). Therefore by the above lemma, if F a F = Fa l F, we have ^jiXi^i)'1 E> I oci1 = X 1 (det(a))^ i %(ai l )g I ai l = %"!(det(a))g | [FaF]. T(q)* = %-1(q)T(q)

Thus we have

if q is prime to N for T(q) on

5k(Fo(N),%). More generally, we have T(n)* = %-1(n)T(n) if n is prime to N.

(10a) Note that x =

r

normalizes Fo(pr) and for a = L

, xax

= a1.

Thus TfCFaFlT" 1 = F a l F and # ( F \ F a l F ) = #(F\FaF) for F = F 0 (p r ). Therefore we can choose the decomposition F a F = IIJFOCJ so that F a l F = IIjFaj 1 is a disjoint union. Since FocT = LIjFxajT"1 is also a disjoint union, by (lb) and x(Oj) = xCCiOjT'1)1), we have

f I T*(q) = Xj

Xj

^

We write Tx(q) for T(q) on 5k(Fo(pr)5X)- Then for any f e 5k(F0(pr),X) Thus we have (10b)

xT^(q)x 4 = T*(q) = T(q)* for all primes q.

Theorem 2 (Hecke). If either % is primitive modulo p r or r = 0, then Hk(ro(pr)>X;C) is semi-simple and ^4(ro(pr)5X) ^ spanned by common eigenforms f of all Hecke operators T(n) such that f | T(n) = a(n,f)f.

146

5: Elliptic modular forms and their L-functions

Although this fact is true for all k > 1, all Fo(N), and all % primitive modulo arbitrary N, we prove the result only when k > 2 and N = p r . (See [M, Th.4.7.2, Th.7.2.18] for the proof in the general case). Proof. First we prove the theorem for SL2(Z). In this case, we see from (10) that (f|T(n),g) = (f,g|T(n)) and thus T(n) is a hermitian operator. Since the T(n)'s are mutually commutative, we can diagonalize T(n) simultaneously on 5k(F). This shows that hk(SL2(Z);C) can be embedded into a product of copies of C and hence hk(SL2(Z);C) is itself a product of copies of C. Note that h k (SL 2 (Z);C) = C r for r = dim c 5k(SL 2 (Z)) because Hom c (h k (F;C),C) = 5k(F). Then each projection X[ of hk(SL 2 (Z);C) into C is a C-algebra homomorphism, and {X\, ..., XT} form a basis of Homc(hk(F;C),C). Let X be one of the A,i's and f be the corresponding element in 5k(F). Then a(n,f) = = A,(T(n)) and hence f = lT_X(T(n))qn. Moreover a(m,f | T(n)) = (T(m),f | T(n)> = a(l,f | T(n)T(m)) = = ^(T(m)T(n)) = X(T(n))a(m,f). Thus f I T(n) = ^(T(n))f and f is a common eigenform of all Hecke operators T(n) belonging to the algebra homomorphism X. We write £ = X°° ^i(T(n))qn. n=l

Then {fj}i gives a basis of 5k(F) over C. Now we show that Ek is a common eigenform of all Hecke operators. We show more generally that Ek(X) I T(q) = ak.if5c(q)Ek(x) for every prime q. We thus compute a(n,Ek(%) I T(q)) = a(nq,Ek(x))+x(q)qk-1a(n/q,Ek(%)). If n is prime to q, then °k-i, x ( n q) = X d l n q Z W ^ ' 1 = Xd|nZb| q X(bd)(bd) k - 1 = a k .i i% (n)o k .i, % (q). Therefore a(n,Ek(%) | T(q)) = ak.i>x(q)a(n,Ek(x)). If n is divisible by q, then a(n,Ek(%) | T(q)) = a(nq,Ek(%))+%(q)qk-1a(n/q,Ek(%))

= X d i n q %( d ) dk " 1+ %(q) c i k " 1 X

On the other hand, we have

5.3. Hecke operators In fact, obviously,

{d | d | n q } = {d | d|n}U{qb

147 b | n } . The intersection

d | n}f|{qb | b | n} is given by {d I d I n, d = qb, b | n} = {qb | b | (n/q)}. Similarly we can show Gk(%) I T(n) = a'k-i,x(n)Gk(%). Anyway this shows that fA4-(SL2(Z)) has a basis {fi, •••, fr-i, Ek} which consists of common eigenforms of all Hecke operators T(n), and hence the desired assertion for f*4(SL2(Z)) follows from this. Now we consider 5k(To(pr)>%)- By (10), -\/%(n) T(n) is self-adjoint if n is prime to p. Thus, we can find a basis of 5k(Fo(pr)>%) consisting of common eigenforms of T(n) for all n prime to p. We shall show that if f ^ 0 is a common eigenform of all operators T(n) for n prime to p and if % is primitive modulo p r , then f is an eigenform even for T(p). We consider the following set of positive integers: X = {n | pjn, a(n,f)*0}. We first show that X is not empty. If this set is empty, then a(n,f) * 0 only if n is divisible by p. Let a = L. and put \9 VJ I fa b^i fa M 1 Then g satisfies g I A = %(d)g for G O = a T o ( p ) a and is also (1 1^ ifa b^ fa pb^ invariant under U = I Q J . Note that a , a = , , and O = {

(a

b^ I

V

J

, M b € p Z , c e p r Z , a,dG Z ,

ad-bc = 1 } .

To show that g = 0, we take integers m and n so that ( m p ^ + l X n p ^ + l ) = 1 mod p r but %(rvpx~l+l) * 1. Note that r-l

and thus gl Y= JcCnp^+lJg but at the same time g | u m = g, g r u ^ ' ^ g and g I u n = g. Thus %(np r " 1 +l)g = g and hence g = 0. This shows that if f ** 0, X is not empty. Take any n in X. Then writing f | T(n) = a(n)f, we have 0 * a(n,f) = (T(n),f) = a(l,f I T(n)) = a(n)a(l,f). Thus a(l,f) ^ 0 for any non-zero common eigenform f of Hecke operators T(n) for n prime to p. Since g = f I T(p) is also a common eigenform of all T(n) for n prime to p, a(l,g)*0. Put h = a(l,g)f - a(l,f)g. By definition,

148

5: Elliptic modular forms and their L-functions a(l,h) = a(l,g)a(l,f)-a(l,f)a(l,g) = 0.

Since h is again a common eigenform of all T(n) for n prime to p, and hence a ( l , h ) ^ 0 if h * 0 . Thus h must be 0 and f|T(p) is a constant multiple of f; namely, f is an eigenform of T(p). Thus 5k(Fo(pr)>%) has a basis consisting of common eigenforms of all Hecke operators. Next, we shall show that ^k(Fo(pr)>%) is spanned by 5k(Fo(pr)>%) and Ek(z,%) and Gk(z,%). Since f^(SL2(Z)) = {0} (Theorem 2.1), we may assume either k > 2 or % ^ 1 and k > 2. Consider the sequence

o -> 5k(r0(Pr),%) -» ^ k O W X x ) - 2 - > c 2 -> o.

(*)

The last map

Q

I. The

surjectivity of (p follows from the fact that (p(Ek(%)) = (2"1L(l-k,%),0) and (p(Gk(%)) = (O.w'LCl-kjX"1)) with w' ^ 0, which is a consequence of Ek(z;%) I k8 = w G k ^ x " 1 ) with w * 0 (see Proposition 1.1). P Since by the functional equation (Theorem 2.3.2), L(l-k,%) is a non-zero constant multiple of L(k,%"1), which is non-zero because the Euler product converges if k > 1. Thus (p is surjective. Now let us show the exactness of the middle term. Pick a modular form f in fA4(Fo(pr),%) and suppose that cp(f) = 0. By the strong approximation theorem (see Lemma 6.1.1), for each element a of SL2(Z), we can find y=\ (i)

a = y8

(a

b\

e Fo(p ) so that either

for some integer j in the interval [l,p r ],

or l

°l y ( s. °"l A for some integer i in the interval [l,prr"ss] and s in [l,r]. p" Then, in case (i), a(0,f | a ) = %(d)a(O,f | 8) = 0. Now we deal with case (ii). Write simply F = Ti(p r ) and Y = F 0 (p r )- Then we see that T / F = (Z/p r Z) x (ii)

via

. H d mod p r , and Y/Y

acts on equivalence classes of cusps of F :

C = F\SL 2 (Z)/Foo, where F o o = | ± H

™1 e T ' l Letting SL 2 (Z) act on the

r 0 (A column vectors (Z/pZ) , we see that F is the stabilizer of the vector . Thus we can identify C with F\(Z/p Z) . Then the cusp corresponding to for i prime to p is given by the vector xi =

s.

s.

. Then the orbit O(x0 of

5.3. Hecke operators

149

under the action of (Z/prZ)x = F/F' is isomorphic to (Z/pr"sZ)x via (Z/pr"sZ)x B i h-> Xj. Considering the function c[> : i I—> a(0,f I

S.

) as a function on

(Z/p r - s Z) x , we see that <|>(di) = %(d) 0 and % is primitive modulo p r , we can find d = l+jpr"s such that %(d) * 1. Since the action of (Z/p r Z)x on O(XJ) factors through (Z/pr"sZ)x, x(d)<>(i) = <|>(di) = <|>(i). This implies $ vanishes identically. That is, for the cusp in case (ii), the constant term of all modular forms in ^k(ro(p r )>%) vanishes. This shows that Ker(cp) = 5 k (r 0 (p r ),%). Thus af k (r 0 (p r ),X) is spanned by 5k(Fo(pr),X) and Ek(z,%) and Gk(z,%). We have already verified that Ek(X) I T ( n ) = ^k-i, x( n) E k(Z) and Gk(%) I T(n) = a'k-i,x(n) Gk(X) for all n. This finishes the proof of Theorem 2. We record the following fact shown in the proof of Theorem 2. Corollary 1. Suppose that % is primitive modulo p r and let f and g be elements in 5k(ro(pr)>%)« Then if f is a common eigenfunction of T(n) for all n prime to p, then f is also an eigenfunction of T(p). If f and g have the same eigenvalues for T(n) for all n prime to p and g * 0, then f is a constant multiple of g. Corollary 2. Let f be a common eigenform of all Hecke operators with a(l,f) = 1 in 5k(SL2(Z)). Then the field Q(f) generated by all the Fourier coefficients of f is a finite extension of Q. Moreover a(n,f) are all algebraic integers and for all ce Gal(Q/Q), f° is again a common eigenform in 5k(SL2(Z)).

Proof. Let F = SL2(Z). We know that f = a = S°° k(T(n))qn for an algen=l

bra homomorphism X : hk(F,%;C) = hk(r,%;Q)®QC —» C. Thus X induces a Q-algebra homomorphism of hk(F;Q) into C. Note that A,(hk(F;Q)) is an algebra of finite dimension over Q which is generated by X(T(n)). Thus Q(f) = ^(hk(r;Q)) and Q(f) is a finite extension over Q. The image X(h^(T;Z)) is contained in the integer ring of A,(hk(F;Q)). In fact, by representing T(n) as a matrix using the basis hi in Theorem 2.1, we see that (ho I T ( n ) , - - , h r _i|T(n)) = (ho,---,hr_i)A(n) with A(n) e Mr (Z). Thus A,(T(n)) is a characteristic root of integral matrix A(n) which is an algebraic integer. We know that HomZ-aig(hk(F,Z),Q) s {f | f I T(n) = ?i(T(n))f, a(l,f) = 1}: X H> fx Naturally Gal(Q/Q) acts on the left-hand side. The action is interpreted into f h-> f° on the right-hand side, which shows the last assertion.

150 5: Elliptic modular forms and their L-functions We will see the assertion of Corollary 2 also holds for 5k(Fo(pr),%) for primitive % in the following section.

§5.4. The Petersson inner product and the Rankin product In this section, we study how we can explicitly construct a basis for the spaces of modular forms (as we have just done for F = SL2(Z)). Theorem 1. Let a > 2 be an integer and let \|/ be a character modulo p r with A|/(-l) = (-l)a- Suppose k>2a+2. Then for any primitive character % modulo p r ( r > l ) with %(-l) = (-l) k , there exist finitely many positive integers n\, n2, •••, nr such that G^Y^k-^X) I T ( n i) for i = 1, ---,r together with Gk(%) and Ek(%) span ^4(ro(pr),%) over C. The above assertion is true with Eaty"1) in place of G a (\|/ 4 ). Similarly

f?Vfk(SL2(Z)) is spanned by E k and E a E k _ a |T(ni)

for some n{ and 4 < a e 2Z such that k > 2a+2. A more general result is obtained in [Wil] (for example, the assertion of the theorem is true even if a = 1). The proof we will give later is basically the same as in [Wil] although it is a little simpler because of our assumption that a > 2. For each Dirichlet character %, we write Z[%] for the subring of C generated by all the values of %. Before proving the theorem, we list several corollaries: Corollary 1 (duality). If k > 6 and F = Fo(p r p) and % is a primitive character modulo p r p / < 9 r r > 0 , then #4(F,%;A) = f&4(r,%;Z|xl)®z[x]A for any Z[x\-subalgebra A of C, where p = 4 or p according as p = 2 or not. Moreover under the pairing in Theorem 3.1, we have the following isomorphisms: HomA(Hk(F,%;A),A) = /nk(F,x;A), Hom A (m k (r,z;A),A) = Hk(F,%;A), Hom A (h k (F,x;A),A) = 5k(F,%;A), HomA(5k(F,x;A),A) = hk(F,%;A). Proof. By the theorem, we can find a basis {fih=i r of ^k(r,%) in ^ie(r,%;Z[%]). In fact, we take any character \\f modulo pp r with \|/(-l) = -l. Then taking a in the theorem to be 2, we find {fi}i=i,...,r among Gi(\\f-l)Ek.1(\\fx)\T(ni), Gk(%) and Ek(%) which form a basis of af k (I\x;K) over K, where K is the field Q(%,\|/) generated by the values of % and \|/. Then choosing a suitable element a* e K for each i, we can easily show that the fi = Tr(aif i) = Saai a fi' a give a basis desired with coefficients in Z[%], where a runs over Gal(K/Q(%)). Thus the natural linear map from ^4(r,%;Z[%])®z[%]C! into #4CT,%) is surjective. By our construction of fi we can find ni, ..., nr so that det(a(ni,fj))ij=i r ^ 0. Then for any (() G fA4(F,%;Q[%]), we can solve simultaneously the linear equations £jXja(ni,fj) = a(ni,<|)) (i = l,...,r) within

5.4. The Petersson inner product and the Rankin product 151 Q(%). Then we see that <>| is a linear combination of £ with coefficients in Q(%). Thus dimQ(x)fA4(r,%;Q[x]) < dim c (^k(r,%)). This shows fM"k(r,%;Z[%])<8)C = f^k(r,%) and mk(r,%;Z[%])®C = ^fk(r,%). In the same manner as in the proof of Theorem 3.1, we know that s H k (T,x;Z[x]), Hom ZW (H k (r,x;Z[x]),Z[x]) = «k(T,x;Z[x]). Obviously by definition, Hk(T,x;A) is a surjective image of Hk(F,%;Z[%])(8)A. If the image of an element T in Hk(r,%;Z[%])®A vanishes in Hk(r,%;A), extending scalar to C, it vanishes in H k (I\x;C). Since a4(T,x) = *4(T,x;Z[xl)®C, we know that T = 0. Thus Hk(r,%;Z[%])®A = Hk(T,x;A). Then we have wk(r,x;Z[x])®z[x]A s HomZ[X](Hk(r,5c;Z[x]),Z[%])(8)A = Hom A (H k (r,%;A),A) = /%(r,%;A). This shows the assertion for A. Now we note a byproduct of the above argument. We have an exact sequence 0 -> ftf k (I\x;A) -> mk(r,%;A) -> N(A) -> 0 for a torsion A-module N. Since A is flat over Z[%], by tensoring with A, we have another exact sequence 0 -> *4O\x;Z[3c])A -> mk(r,%;Z[%])(8)A -+ N(Z[xl)®A -> 0. Since the middle terms of the above two sequences coincide and the last maps of these sequences also coincide under this identification, we have

;A) = *4(r,x;Z[x])®A. By Corollary 1 and Theorem 1, we know that hk(r0(ppr),%;C) = hk(r0(ppr),z;Q(x))®Q(%)c if k > 4 . Then, similarly to the proof of Corollary 3.1, we know the following result when k > 6 . The result is actually true for all k > 0 [Sh3, p.789]. We will later show this result for k > 2 as Theorem 6.3.2.

Corollary 2. Let f be a common eigenform of all Hecke operators in ^k(ro(ppr),%) such that a(l,f) = 1. Then the field Q(f) generated by all the Fourier coefficients of f is a finite extension of Q. Moreover a(n,f) are all algebraic integers and for all c e Gal(Q/Q), f° is again a common eigenform in

To prove the theorem, we shall prepare with some lemmas.

152 5: Elliptic modular forms and their L-functions Lemma 1. Let g be an element of 5k(F,%) and {f} be the basis consisting of common eigenforms. We assume that F = Fo(ppr) if X is primitive modulo p p r and F = SL2(Z) if % is trivial. If we write g = Zfc(f,g)f with c(f,g) e C, then c(f,g) = (g,f)/(f,f), a«^ */ c(f,g) * 0 /or a// f, then g | T(ni) for some n* spara 5k(F,%). Proof. Since the proof is exactly the same for SL2(Z) and Fo(pr), we only treat the case of F = Fo(ppr). As already seen, hk(F,%;C) is semi-simple and hence is isomorphic to C d for some d as an algebra. Let X[ be the i-th projection of hk(F,%;C) onto C, which is an algebra homomorphism. Then X{ spans Homc(hk(F,%;C),C) = 5k(F,%). Let fi be the cusp form corresponding to X{. Then {fi} forms a basis of 5k(F,%). We see that a(n,fi) = (T(n),fi) = ^(T(n)) (in particular, a(l,fi) = 1) and a(m,fi I T(n)) = = a(l,fi I T(m)T(n)) = Xi(T(m)T(n)) = Ad(T(n))a(m,fi). Therefore we have fi | T(n) = A,i(T(n))fi. If i * j , then we can find T(n) such that A,i(T(n)) ^ \j(T(n)) since the T(n)'s generate the Hecke algebra hk(F,%;C). Thus the chosen basis coincides with {£}. In the proof of the semisimplicity of the Hecke algebras, we have shown that if f is a non-trivial common eigenform of Hecke operators T(n) for all n prime to p, then f is also an eigenform for T(p) and a(l,f) ^ 0. Moreover we have shown that T(n)* = %(nYlT(n) for n prime to p, where T(n)* is the adjoint of T(n) under the Petersson inner product. Note that T(n)* commutes with T(p)*. Therefore T(n) commutes with T(p)* if n is prime to p. If f is a common eigenform for all T(n), f | T(p)* is a common eigenform for T(n) for all n prime to p having the same eigenvalues as f. Therefore if f | T(p)* * 0, then a ( l , f | T ( p ) * ) * 0 and h = a(l,f I T(p)*)f - a(l,f)f I T(p)* is a common eigenform of T(n) for all n prime to p with a(l,h) = 0; hence h = 0. Hence f is also an eigenform of T(p)*. Thus what we have shown is that if f is a common eigenform for T(n) with n prime to p, then f is also a common eigenform of T(n) and T(n)* for all n including p. If there are two non-trivial eigenforms f and g belonging to the same eigenvalues of T(n) for all n prime to p, then again by putting h = a(l,f)g - a(l,g)f, we know that h is a common eigenform of T(n) for all n prime to p with a(l,h) = 0. Thus h = 0 and g is a constant multiple of f. This shows that ^i is determined by the value at T(n) for n prime to p. Thus, if i ^ j , then we can find n prime to p such that Xi(T(n)) * Xj(T(n)). Since T(n)* =x(n)- 1 T(n) > T = ( ^ c O O ^ K n ) is selfadjoint and (V3C(n))"1^i(T(n))(fi,fj) = (fi I T,fj) = (fi5fj | T) = (^X(n) )"Vj(T(n))(fi,fj).

5.4. The Petersson inner product and the Rankin product 153 This shows that (fi,fj) = 0 if i ^ j . Therefore, writing g = Ei c(fi,g)fi, then (g,fi) = c(fi,g)(fi,fi) and the formula c(fi,g) = (g,fi)/(fi,fi) follows. Let us now prove the last assertion of the lemma. Let M be the vector subspace of hk(F,%;C) generated by T(n) for all n. Then the pairing (h,f) = a(l,f |h) is still non-degenerate on M (in fact, if (h,f)=O for all h e M, a(n,f) = (T(n),f) = 0 and f = 0). Thus dim c (h k (r,x;C)) > dimcM > dim c (5 k (F,x)) = dim c (h k (F,x;C)). Therefore hk(F,%;C) = M. (This fact is even true for any subring A of C containing Z[%]: hk(F,%;A) = Z n AT(n).) Let li be the idempotent corresponding to the i-th factor C of hk(F,%;C); thus A,j(li) = 8ij. Then one can express li = ZjC(nij)T(riij) with c(nij) e C and njj > 0. Then for g as in the lemma, we have j

On the other hand, g I li = Ejc(nij)g | T(nij). Thus if c(fi,g) * 0, then £ can be expressed as a linear combination of g | T(njj). By picking a basis out of {g | T(nij)}ij, we get a desired basis g I T(ni). To prove Theorem 1, we compute for each common eigenform f, c(g,f) = (g,f)/(f,f) for g = Gaty'^Ek-atyJC). Note that if a> 2, G&(\\fA) has a non-trivial constant term only at the cusp 0 and Ek.a(\|/%) has a non-trivial constant term only at infinity. Thus g is a cusp form (if a > 2). We write £ for \|/%, / for k-a and h for Gaty"1). Then by definition

where (cpr,d) runs over all pairs of relatively prime integers with d > 0. Since (cpr,d) is relatively prime, we can find integers a and b such that ad-bcpr = 1 and y = C

r

e ro(pr)« If we pick another 5 = c

V P ^y vP * * the same lower row, then y8 = L e Foo, where

r . = {ye r o ( P r ) I y(-) = « j = {±^ ™j | Therefore, we know that Thus at least formally

in Fo(pr) with

r

"• J

m

154

5: Elliptic modular forms and their L-functions c(g,f)(f,f) =

where <& is the fundamental domain of Fo(pr). Note that Y*(y"2dxAdy) = y"2dxAdy for all y e SL 2 (R), y(y(z))k = y(z) k | j(y,z) I "2k, and fC/(i))h(Y(z)) = W)Kz%~\i) |j(Y,z) | 2kj(Y,z)-'. Therefore, we have f(i)h(Z)^1(Y)j(Y,z)"/ y(z)k = f(Y(zl)h(Y(z))y(Y(z))k and for a non-zero constant Co (see §1) c(g,f)(f,f) = Since L L g r \r (pVY*^ *s

a

fundamental domain of r «

and the domain

{z = x + V - I y I y > 0 and 0 < x < 1} also gives the fundamental domain of F^, we have, for e(z) = exp(2rc V-Tz), c(g,f)(f,f) = coLC/,^1) J ^

fh(z)yk-2dxdy

= coLC/,^1) J~ ^ m n a(m,f) c a(n,h)e((m+n)V z ry)£ e((n-m)x)dxyk-2dy, where c denotes complex conjugation. As is well known, (1 if n = m, ri e((n-m)x)dx = \~ . ' ' [0 otherwise. Jo v Therefore, oo

c(g,f)(f,f) = coLC/,^1) J ~ £

a(n,f)ca(n,h)exp(-43my)yk-2dy

n=l

= c o (47i)-T(k-l)L(/,^ 1 ) £

a(n,f)ca(n,h)n-s | s=k .!.

n=l

By the formula before Proposition 1.1, we know that en - N k -'

(k M)!

'

We need to justify that we may interchange the summation and the integral in the above computation. By (3.8b), we have I a(n,f) I < Cn1^2 and by Proposition 1.2 and (1.4a,b), la(n,h)| < C'na"1+e with any e > 0 for constants C and C independent of n; therefore, the computation will be justified if k-1 > 2 + a + e, that is, k > 2a+2+2e. We now define the Rankin product zeta function of f and h by (2)

D(s,f,h) = ]T a(n,f)ca(n,h)n"s. n=l

5.4. The Petersson inner product and the Rankin product 155 1 Here h is a general modular form, not necessarily GaCij/" ). If f and h are cusp forms, then | a(n,f)ca(n,h) | < Cn (k+a)/2 and thus D(s,f,h) converges absolutely if Re(s)> 1 + ^ r because D(s,f,h) is dominated by £(s-(k+a)/2). If h is not cuspidal, then | a(n,f)ca(n,h) | < Cn(k/2)+a"1 and D(s,f,h) converges absolutely if Re(s) > a+2 by the same reasoning. Thus Theorem 1 follows from Lemma 2. For two common eigenforms h e ^&(To(ppT),\\f'1) f e 5 k (r 0 (pp r ),X), D(k-l,f,h) * 0 if k > 2a+2.

and

Proof. We compute the Euler product of D(s,f,h). We note the recurrence relation for each prime q: T(qr)T(q) = T(q r+1 ) + X(q)qk'1T(qr-1). Therefore for any common eigenforms f and h of all Hecke operators such that a(l,f) = a(l,h) = 1, we already know that f | T(n) = a(n,f)f and h | T(n) = a(n,h)h. For simplicity, we write a(n) = a(n,f)c and b(n) = a(n,h). From the relation, we know that a(qr)a(q) = a(qr+1) + xCqjV'Vq'" 1 ) and b(qr)b(q) = b(q r+1 ) +X|/(q)"1qk-1b(qr-1) if r > l , where \|/ (resp. %) is the character of h (resp. f). We take two roots a, (3 of X2-a(q)X+%"1(q)qk'1 = 0 and a',p' of X2-b(q)X+\(/(q)-1qk-1 = 0. We write formally P(X) = S°° a(qn)Xn. Then we have n=0 oo

P(X)(cc+p) = ^

oo

a(q

n+1

) X n + a p ^ a(q n-1 )X n = (P(X)-l)X- 1 +apXP(X).

n=0

n=l

This shows that

P(X) = (l-(a+p)X+apX2)"1 = (l-aX)-1(l-pX)"1 = ^

—

and a(qn) = (a-fiy1^1-^*1),

b(qn) = (a'-pT^oc'^-p' 11 * 1 ),

Now computing V 1 /_. n+l n

n

n

>.(a

Q(X) = V a(q )b(q )X = J>=2-

(a-p)(a'-P 1 )Xll-aa'X

on+1 \/_,i n+1

-p

)(a

Otn+lN-vrn

-p

)X

(a-p)(a'-p')

1

l

1-ap'X

1-Pa'X

l +

1-PP'X.

156

5: Elliptic modular forms and their L-functions l-gg'pp'X 2 i-(X¥)' 1 (q)q k+a ' 2 x 2

(l-aa'XXl-ap'XXl-poc'XXl-pp'X)' we find that

This infinite product converges absolutely at s = k-l if k > 2a+2, because, writing A = a or p and B = a' or p \ I ABq"s | < q"1 if Re(s) > l+^+a-l for q sufficiently large, and the Euler product of £(s) converges if Re(s) > 1. Since f°(z) = f(-z), then f° e 5k(F,%c) and (f,f°) = (f,f). Therefore we have, for g = hcEk.a(\|/'1Z)» c(g,f)(f,f) = c o (47c) 1 - k r(k-l)L(2-k-a+2s,x-V)^(s,f,h) I s=k-i = co(47i)1-]T(k-l)L(k-a,%-V)^(k-l,f,h). The convergence of the Euler product gives us the non-vanishing of c(g,f) and we now have Theorem 1. Corollary 3. Suppose that f and h are normalized common eigenforms of all Hecke operators in 5k(Fo(N),%) and fAfa(To(N),\|/), respectively. Put

Suppose that %~l\f is primitive modulo N = p r . If k > 2a+2, then T(k-l,f,h) e Q(f,h), where Q(f,h) is the finite extension of Q generated by the Fourier coefficients of f and h. Moreover for all a e Aut(C), T(k-l,f,h)° = T(k-l,f°,h o ). This is a special case of a general algebraicity theorem of Shimura (see Theorem 10.2.1 and also Theorem 7.4.1). As already seen NL(k-a,%-V)D(k-l,f,h) = T(k-l,f,h). Then the assertion follows easily from this formula and the fact that

We now want to show that f° c = f° a for our later use. The n-th Fourier coefficient a(n,f) of f is the eigenvalue of T(n). On the other hand,

5.5. Standard L-functions of holomorphic modular forms ^ and

157

T(n) is self-adjoint if n is prime to p. Thus (<^%(n)) a(n,f) is real

a(n,f)c = x(n)"1a(n,f). In particular, a(n,f)ca = x°(n)"1a(n,f0) = a(n,f°c). Since the p-th Fourier coefficient of f is determined by those for n prime to p as shown in the proof of Theorem 3.2, we also have a(p,f)ca = a(p,f) ac . In fact fG-f°c is a common a ac eigenform of all T(n) for n prime to p and a(l,f° -f ) = 0 , and hence

The above proof in particular shows that (3)

a(n,f)c = x'^n^Oijf) for n prime to p if f is a normalized eigenform in 5k(ro(pr),X)-

§5.5. Standard L-functions of holomorphic modular forms Let X : Hk(Fo(pr),x;C) —> C be a C-algebra homomorphism. We assume that X is a primitive character modulo N = p r . We define the standard L-function of X by

By Theorem 3.1, we have a common eigenform of all Hecke operators f e ^k(r o (p r ),X) such that a(n,f) = X(T(n)) for all n. We then know from (3.8b) and (1.4a,b) that I A,(T(n))n-s I < Cn" Re(s)+(k/2) if f is a cusp form, and U(T(n))n" s | < Cn- Re(s)+k " 1+e (for all £ > 0) if f is just a modular form. When f is a cusp form, i.e. X factors through hk(ro(pr)>%)> then we see that (2)

L(s,X) converges absolutely if Re(s) > 1+-|.

If f is associated with X, i.e., f = Z°° X(T(n))qn, then by (3.10a,b) n=l

(f \x) I T(n) = ^(T(n))c(f | x) for all n,

(3)

where c denotes complex conjugation. Supposing that x is primitive modulo pr, we know from Corollary 3.2 that f | x is a constant multiple of f°: Proposition

1. Let

f = £°° X,(T(n))qn for an algebra homomorphism n=l

r

X : h k (r 0 (p ),X) -^ C. Define Xc : h k (r 0 (p r ),% c ) -* C by Xc = c°XoC. We suppose that % is primitive modulo pr. Then we have

158

5: Elliptic modular forms and their L-functions

(4)

f | x = pr(k-2)/2W(?i) £ MT(n))cqn, n=l

for a constant W(X) with W(X)W(XC) = %(-l) and I W(A.) I = 1. Here we agree to put Fo(p°) = SL2(Z) and % = id in this case. Proof. We only need to prove the last assertion. By Corollary 3.2, W(X) * 0. Since

T 2 = - p r L °X we see that (f | x)(z) = p (k " 1)r f(-l/p r z))(p r z)- k and

( f | x ) h = %(-l)p(k-2)rf. Thus W(X)W(XC) =x(-l). Applying fM> f(=I) to the formula (4), we see that W(XC) = % ( - l ) W ( ^ ) c . This shows that

I

| = 1.

We now show the analytic continuation of L(s,X) and its functional equation. First assume that f e 5k(Fo(pr)>%) for a Dirichlet character % modulo p r (here % may be imprimitive). In this case, by (3.8a), the integral J f(i'y)ys"1dy is absolutely convergent for all s and gives an entire function of s. By the same type of computation as in §2.2, we have, if Re(s) > l+w,

Thus L(s,X) has an analytic continuation to an entire function on the whole s-plane. On the other hand, if either % is primitive modulo p r or r = 0, we see that f

= i-kP"rJ0

|x(/y)ys-ldy

f ^

This shows Theorem 1. Suppose that % is primitive modulo p r . Then, for each algebra homomorphism X : hk(F()(pr),%) "^ C, the L-function L(s,X) is continued to an entire function on the whole s-plane. If either r = 0 or % is primitive modulo pr, the L-f unction L(s,X) satisfies the following functional equation rc(s)L(s,A,) = p r((k/2) - s) / k W(^)r c (k-s)L(k-s,A, c ) for F c (s) = (2TC)-T(S). For any primitive character \\f modulo N, we can define

Then we see from a similar computation as in (4.1.6c) that

5.5. Standard L-functions of holomorphic modular forms

159

fv = £ \|/(n)a(n,f)qn.

(5)

n=l

For any algebra homomorphism X : hk(Fo(pr)>%) —> C and a primitive character \|/ : (Z/NZ) X -> C x , we define

£ \|/(n)MT(n))n"s. n=l

Thus we have, as a corollary to the proof of Theorem 1, Corollary 1. For each algebra homomorphism X : hk(ro(pr),%) —> C and each primitive character \\f: (Z/NZ)X —> C x , the L-function L(s,?t<8>\|/) is continued to an entire function on the whole s-plane. Proof. Although we proved Theorem 1 assuming that % is primitive modulo p r , it is clear from its proof that J f(iy)ys"1dy is an entire function in s if f is a cusp form. Thus what we need to prove is that fv character £ of Fo(M). It is easy to check that (6)

if y = I

is a cusp form for some

, I e Fo(M) for the least common multiple M = [N,pr] of

fa1 bf>i for Y = I c. d. I e Fo(p r ) with d1 = d mod p r . 1

This shows that

f,lr-G
^

^

r=l

Thus we have f¥ e 5k(Fo(M),%\|/2). Define an algebra homomorphism A,®\|/ : hk(Fo([N,p r ]),X) -> C by f¥ | T(n) = (X®\|f)(T(n))fv (i.e. (X®\|/)(T(n)) = \|/(n)X(T(n))). Then if N is prime to p, we can compute W(X®%) explicitly by using W(X) (see for details [Sh, Prop.3.36], [M, §4.3] and [H5, (5.4-5)]).

Chapter 6. Modular forms and cohomology groups In this chapter, we prove the Eichler-Shimura isomorphism between the space of modular forms and the cohomology group on each modular curve. This fact was first proven by Shimura in 1959 in [Shi] (see also [Sh, VIII]). We shall give two proofs in §6.2 of this isomorphism. The first one is the original proof of Shimura based on the two dimension formulas. One is the formula for the space of cusp forms and the other is for the cohomology group. The other proof makes use of harmonic analysis on the modular curve. After studying the Hecke module structure of modular cohomology groups, in §6.5, we construct the p-adic standard L-function of GL(2)/Q following the method (the so-called "p-adic Mellin transform") of Mazur and Manin in [Mzl], [MTT] and [Mnl,2]. Throughout this chapter, we use without warning the cohomological notation and definition described in Appendix at the end of the book. If the reader is not familiar with cohomology theory, it is recommended to have a look at the appendix first.

§6.1. Cohomology of modular groups In this section, we shall prove the dimension formula of the cohomology group of congruence subgroups of SL2(Z) following [Sh, VIII]. Let F be a congruence subgroup of Slyz(Z) and suppose for simplicity that F is torsion-free. (The general case without assuming the torsion-freeness of F is treated in [Sh, VIII].) If N > 4, Ti(N) is torsion-free. In fact, if ±1 * £ e F satisfies £m = 1 for m > 2, then the absolute value of the sum of two eigenvalues of £ is less than 2. Thus | Tr(C) I = 0 or 1. On the other hand, if £ e Fi(N), Tr(Q = 2 mod N. It is easy to check that it is impossible to have | Tr(^) | < 2 and if N > 4 . Exercise 1. If N > 4, why is Tr(Q = 2 mod N

impossible?

Let R be a commutative ring, and for any left R[F]-module M, we consider the group cohomology group H^I^M) (see Appendix for definition). Here we start from an open Riemann surface Y = T\tf. For each s e P ! (Q) = QU{°°}, we q consider the stabilizer F s of s in F. If s is finite, writing s = 7 as a reduced fraction, we can find integers x and y such that qy-px = 1. Putting fq x\ os = a = e SL2(Z), we have G(©°) = s. This shows that a F s a is |Y 1 m\ I 1 contained in ( ± 1 } X U ( Z ) for U(A) =j i I I m e Af- Thus defining the distance from s by d(z) = Im(a' 1 (z))" 1 , we see that d(z) is well defined on FSW, and we put US)£ = ( z e FSV#| d(z) < e}. For sufficiently small e, Us<£ is naturally embedded into FSV# We identify two cusps if U s , e nU t , e * 0 for

6.1. Cohomology of modular groups

161

any 8 > 0. We see easily that two cusps are the same if and only if y(t) = s for some y E F. Thus the set S of cusps is bijective to F\PSL2(Z)/U. Since T is a congruence subgroup, there is a positive integer N such that ri0 r < N , . l l - j 6 S L 2 < Z , ' rc - dn j " 10 ^i . m o d N K

Then it is obvious that #(S) < (SL2(Z):r(N)), which is finite. In the above proof of the finiteness of S, the following strong approximation theorem is proved implicitly: Lemma 1. Let {Ni)iGN be a sequence of integers such that Ni is a divisor of Ni+i for all i. Then SL2(Z) is dense in lim SL2(Z/NiZ). In particular, writing i

Z = ] l p Z p = lim SL2(Z/N!Z), SL2(Z) is dense in SL 2 (Z). Proof. We need to show that the natural map SL2(Z) —> SL2(Z/NZ) is surjective for any positive integer N. Let V = {v E (Z/NZ)21 the order of v is equal to N), where the order means the order of the subgroup generated by v in the additive group (Z/NZ)2. Then the vector v is represented by and q. Thus we can find x and y in Z a =

E SL 2 (Z), avo = v mod N,

for relatively prime p

so that py-qx = 1 and for

where

vo = L L Thus for any

oco E SL 2 (Z/NZ), we can find a E SL 2 (Z) SO that avo = aovo mod N. That is, ccU(Z/NZ) = oc0U(Z/NZ). The natural map U(Z) -> U(Z/NZ) is obviously surjective, and hence by modifying a from the left by an element of U(Z), we can find a' E SL 2 (Z) such that a 1 = ao mod N. This shows the surjectivity. Now we show that

Y-USGSU S)£

= Yo is compact if {Us,e}SGS do not overlap.

We may assume that T = SL2(Z) because the above space is a finite covering of the space Yo for SL 2 (Z). Since a = L damental domain in the strip y(5(z)) = y/(x2+y2) 2

2

2

for 2

B

in

E SL 2 (Z), we can take its funO{ with

z = x+zy

and

| Re(z) | < TT. 8 = ^

Since

^ j e SL 2 (Z),

x +y > 1 <=> x(8(z)) +y(8(z)) < 1. Starting from a given z = zo e B with x +y < 1, we apply 8 to z and bring 8(z) back in B by tanslation by a power of a. We write this element in B as z\. We can now define a sequence zo, zi,..., z n ,..., by repeating this process of first applying 8 and then translating back into B by a power of a, that is, zn = ocm(8(zn_i)) for a suitable integer m so that zn E B. As soon as we get | zn | 2 = Re(z n ) 2 +Im(z n ) 2 > 1, we stop the

162

6: Modular forms and cohomology groups

process. If the sequence is of infinite length, there is a limit point because the unit disk centered at the origin is compact. Since Im(8(z)) = y/(x2+y2) > Im(z), and the ratio Im(8(z))/Im(z) is given by l/(x2+y2). Thus the limit point has to be on the intersection of B and the boundary of the unit disk. This implies that there exist a positive number h such that a fundamental domain of SL2(Z) can be taken in {z e B | Im(z) > h}. In fact, a standard fundamental domain is given as the intersection of B and the domain outside the open unit disk centered at the origin; see [M, 4.1.2]. Thus y(z) = Im(z) is bounded below independent of z in this fundamental domain, and hence y(z) is bounded below and above in the inverse image Oo of Yo in this fundamental domain. That is, Oo is relatively compact. This implies that Yo is compact. To compactify Y = FV^ we need only compactify Us,e. Since Us,e is isomorphic to Uoo,e, we compactify Uoo,e. Since Ho is a subgroup of U, there exists N e Z such that Z H q = e(-sr) gives an isomorphism of Uoo,e into G m (C) = C x and y(z) -» °o <=> q -> 0 on Uoo,e. We then add the cusp °o to Uoc,e so that e : Uoo,£U{°°} —> C gives an isomorphism onto a neighborhood of 0 in C. We write this compactification as Uoo,e and the corresponding compactification of USt£ as Us>e. Then X = YUS is a compact Riemann surface having Us,e as a neighborhood of s € S. Thus we get X and Y = X-S as in Appendix. We make the triangulation of Yo = Y-Use sUs,e for small e and get a complex £ satisfying the conditions (Tl-3) in Appendix. We may assume that the fundamental domain Oo of Yo in the inverse image of Yo in 9{ consists of simplices of £ (we may need to take out some simplices from £ which overlap in Yo). Let Si be the set of all i-simplices of £ We use the notation defined in Appendix for X and S, especially, H$ denotes the various parabolic cohomology group with respect to S (see Proposition A.I). Proposition 1. For any F-module M, let DM = Zy € r (y-l)M. Then we have H 2 (r,M) = M/DM and H 2 (r,M) = 0. Proof. We choose £ in a manner suitable to our proof. We triangulate X so that by cutting along the curves representing a basis of the homology group of X, we get a polygon of 4g sides, where g is the genus of X. We may assume that all the holes of Yo are inside this polygon. Fixing one vertex qo of this polygon and cutting the polygon from qo to each cusp of X, we have another simply connected polygon Oo with 4g+c sides, where c = #(S) is the number of the cusps of X. We may identify this Oo with a fundamental domain O of P. Thus F is generated by 2g+c elements (Xi, Pi, ..., ocg, p g and %\, ..., nc with the sole relation

6.1. Cohomology of modular groups

163

Here we have identified S naturally with {l,2,...,c}. We use the same symbol OQ for the sum of elements in the set S2 of 2-simplices in Oo which represents the domain Oo. Then, using the notation introduced in Appendix

X S ts + XJ Here Sj (resp. s j , ts) denotes the face of Oo corresponding to Oj (resp. Pj, 7is). We extend £ to a triangulation K of fH$ = Tr^Yo) for the projection % : J{ —> Y (as described in Appendix). Thus we have the chain complex: (Aj,9) associated to K. Let u be a cocycle having values in M, i.e., u G A2(M) = HoiriR[r](A2,M) with fo3 =0. Since A2 is generated over R[F] by S2, u is determined by the values of the simplices of S2. Note that ocp-1 = (oc-l)(P-l)+(a-l)+(P-l). Since T is generated by CCJ, pj and TCC, D M

= Eses (^s-l)M + X- =1 {(a r l)M+(pj-l)M}.

If u = 3w with 3w e BP(K,M) (see Proposition A.I for B P ), then u(O 0 ) = X S G S w^s) + X-=i {(aj-Dw(Sj) + (pj-l)w(s t j )} E DM, because w(tc) e (TCC-1)M by the definition of Bp(K,M). Thus u H u(Oo) gives a morphism of HP(K,M) into M/DM. This map is surjective because one can assign an arbitrary value to u(
u(a)+u(b) = £ s e A w(s) + £ s . e B w(s'),

164

6: Modular forms and cohomology groups

w(t) is well defined. Thus we have a parabolic chain w on Oo such that w°3 = u. Then we extend w to Ai by the R[F]-linearity to have w°3 = u on Ai. Thus the map HP(K,M) -» M/DM given by the evaluation at Oo is injective. This finishes the proof of the first assertion. As for the second assertion, we consider the boundary exact sequence (Proposition A.2)

e s€S H 1 (r Wg ,M) -» H?(r,M) -> H2(r,M) -» 0 (exact) \\i ®sesM/(7Cs-l)M

m > M/DM.

The above diagram is commutative by definition, and the second horizontal arrow is the natural projection © S eS m s mod (TCS-1)M H> Z s m s mod DM, which is obviously surjective. Then the exactness of the first row shows the vanishing of H 2 (I\M). By the above proposition, we know that HC(Y,A) = A via the evaluation of 2-cocycles at Oo- As is well known, the Poincare duality between H2(Y,A) and H2(Y,9Y; A) for A = R or C is given by the integration over 2-cycles of closed differential 2-forms, if one computes the cohomology groups using de Rham resolution. Thus we have Corollary 1. We have H 2 (Y,A) = A for all A, and if A = R or

C,this

isomorphism is given by: HC(Y,A) 3 [co] f-> J Yco E A, where co is a closed 2-form representing the de Rham cohomology class [co]. Proposition 2 (dimension formula). Suppose that R is a field and M is of finite dimension over R. Let g be the genus of X. Then we have dim(H1P(r,M)) = (2g-2)dim(M)+dim(H°(r,M))+dim(H^(r,M))+ ^s6Sdim((7Cs-l)M)). Proof. Let ®o be the fundamental domain of Yo as in the proof of Proposition 1. Then S2, Si- {ts I SG S) and So give a triangulation of Yo. Thus by Euler's formula, we have #(S 2 )-#(S r {t s I s e S})+#(S0) = 2-2g, where #(X) denotes the number of elements in a finite set X. On the other hand, #(S0 = rankR[r](A0, and thus dimR(Ai(M)) = dimR(HomR[r](Ai,M)) = #(Si)dimR(M). Note that if we put Ap(M) = ( U G Ai(M) I u(ts) e (JC S -1)M}, then dimR(AXP(M)) = #(Si)dimR(M) - X s because of the exact sequence

6.1. Cohomology of modular groups

165

0 -» Aj,(M) -> Ai(M) -> 0 seS M/(7C s -l)M -> 0. By definition, we have Hj,(K,M) = Zp(K,M)/B1(K,M), H|(K,M) = A 2 ( M ) / B £ ( K , M ) . Thus, writing d(i) (resp. d, d1) for the dimension of the i-th parabolic cohomology group (resp. dim(M), ZSES dimR((7Cc-l)M)), we see that d(0) = dim(Ker(3 : A 0 (M) -> Aj»(M))) = #(S0)d-dim(B1(K,M)), d(l) = dim(Z1P(K,M))-dim(B1(K,M)), d(2) = dim(A2(M))-dim(B|(K,M)) = #(S2)d-(#(Si)d+dt-#(S)d-dim(Z1P(K,M))). Thus we have d(0)-d(l)+d(2) = #(S0)d-(#(Si)d+d!-#(S)d)+#(S2)d = (2-2g)d+d\ That is, d(l) = (2g-2)d+d(0)+d(2)-df, which is the desired formula. We compute more explicitly the dimension of cohomology groups for some special F-modules. For a commutative ring R with identity, we consider the space L(n;R) of homogeneous polynomials of degree n with two indeterminates X and Y. By definition, L(n;R) = L(n;Z)®zR. We define the action of the semi-group M2(R) on L(n;R) by

where yl = Tr(y)-Y= dst(y)Yl. Note that i : M2(R) -^ M2(R) is an involution, i.e., (ocp)1 = p l a l for all a, P e M2(R). Over Q, this representation of M2(Q) on L(n;Q) is equivalent to the symmetric n-th tensor representation of the natural identification of M2(Q) with itself. To make the dimension formula in Proposition 2 computable, we need to compute diiriR(M), dimR(H°(F,M)), dirriR(Hp(r,M)), dimR((7ts-l)M) for M = L(n;R). One can easily compute these dimensions when R is a field K of characteristic 0. We know from the definition that dirriR(M) = n+l (i.e. a standard basis is given by Xn, Xn"xY, ..., Yn). We claim that n o rl if n = 0, (1) dimR(Hu(F,M)) = dim R (Hp(F,M)) = j which follows from Lemma 2 (irreducibility). The F-module L(n;K) is absolutely irreducible if K is afield of characteristic 0 and F is a subgroup of finite index of SL2(Z). Proof. By the density of SL2(Z) in SL 2 (Z p ), we may assume that K = Q p , the algebraic closure of Q p . In fact, V =L(n;Q p ) = L(n;Q)® Q p is irreducible if and only if L(n;Q) is irreducible. Suppose that we have a SL2(Zp)-stable vector subspace W # {0} in V. We pick an element P(X,Y) in W whose degree j with respect to Y is maximal. Since u(Xn^YJ)

166

6: Modular forms and cohomology groups

= (X+Y)n"jYJ for u =

, j has to be equal to n. Since oca = I * _}

acts on X^11"1 by the scalar a11"21, writing P(X,Y) = liCiX^ 11 ' 1 with c 0 * 0, we see that a a P = LiCia 1 1 " 2 ^^ 1 1 " 1 e W. Choosing distinct ai for i = 0, ...,n, we can write Y n as a linear combination of ocaiP; that is, Y n e W. Note that SiYn = (X+Y) n e W for the transpose lu of u. Since (X+Y)n involves ICY1'1 non-trivially for all i, ICY1'1 is a linear combination of {a ai (X+Y) n )i c W. This shows that W 3 V, which finishes the proof. To compute dimK((7is-l)M) when n > 0, we insert a definition. A cusp s is called a regular cusp if G ^ I ^ G S is contained in U(Z). A cusp which is not regular is called an irregular cusp. Then, we have, for some 0 * h e Z _i _ f u h if s is regular, l-u if s is irregular. We see that M/(7TS-1)M = M/(±u h -l)M via P h ^ G ^ P . If either s is regular or n is even and positive, we see easily that M/(u h -l)M = K by P h-» P(l,0), because -1 acts trivially on M. Then (2a)

If either n > 0 is even or s is regular, dirriK((7ts-l)L(n;K)) = n.

Now we treat the remaining case where n is odd and s is irregular. In this case, ds'^s^s-l is invertible on L(n;R) unless R is of characteristic 2. Thus (2b)

If n is odd and s is irregular, dirriK((7vl)M) = n+1.

Summing up the above formulas, we have Corollary 2. If T is torsion-free and K is a field of characteristic 0, then (3)

dim K (H 1 P (r,L n (K))) = j(2g-2)(n+l)+n#(S)+5 n #(Si), if n > 0, 1 2g, // n = 0,

where Si is the subset of S consisting of irregular cusps and 8n is 0 or 1 according as n is even or odd. Remark 1. If the reader is familiar with the dimension formula of the space of cusp forms 5k(O, he will notice the curious identity (4)

dim R (5 k (r)) = dimKH1P(r>L(n;K)).

Since the dimension formula of Sn+i^X) is proven in various places (for example [Sh, Theorems 2.24 and 2.25] and [M, §2.5]) and since its direct proof without using cohomology groups requires either the Riemann-Roch theorem of curves or

6.2. Eichler-Shimura isomorphisms

167

the Eichler-Selberg trace formula, we shall not give the direct proof here. In the following section, we prove the Eichler-Shimura isomorphism 5k(F) = Hp(T,L(n;R)) which gives an indirect proof of this fact. In fact, we will give two proofs of this fact, and the first one actually uses the dimension formula (4).

§6.2. Eichler-Shimura isomorphisms Now we establish a canonical R-linear isomorphism between Sn+2(T) and the cohomology group Hp(F,L(n;R)). We give two proofs. First we repeat the proof given in [Sh, VIII] which is based on the dimension formulas for 5k+2(O and Hp(F,L(n;R)), and the second one is based on the Hodge theory of manifolds with boundary. For each point

z e fH, we consider the L(n;C)-valued differential form n

8 n (z) = (X-zY) dz. Let e = ( ^

l

\

Then for y = (*

can easily check (noting foeoc = det(a)e for a e GL2(C))

= f(X Thus, putting co(f) = 2n V = 4f(z)8 n (z) for f e 5 n +2(F), we have y*o(f) = yco(f); thus co(f) is a section of the sheaf of differential forms having values in the locally constant sheaf £( n ;C) associated to the F-module L(n;C). Referring the details of L(n;C) to Appendix, let us briefly state the definition of L(n;C) = L(n;C). For any F-module M, we can define T = IV£<M letting F act on JHXM by y(z,m) = (y(z),ym). Then M is the sheaf of locally constant sections of the projection n : T -> Y = FV# Here we take L(n;R) as M. We fix one point z in #"* = ^ U P ^ Q ) (which is naturally embedded in P^C)) and consider the integral, for each y e F,

q>z(f)(Y) = JJ (Z) Re(co(f)) e L(n;R). Since co(f) is holomorphic, the integral is independent of the choice of the path between z and y(z). When z e P ! (Q) (i.e. a cusp), we suppose that the projected image of the path in X is a well defined path near the cusp z e S . Then the integral is convergent even if z is a cusp, because the cusp form is decreasing exponentially towards the cusp. For another point z1 e #*, we have


168

6: Modular forms and cohomology groups

and for y,5 e F (pz(f)(Y§) = JzY5(Z)Re(co(f)) = J^z()Z)Re(co(f))+9z(f)(y)==79z(f)(5)+9z^ Thus (pz(f) is a 1-cocycle with values in L(n;R) whose cohomology class is independent of the choice of the base point z. If 7i(s) = s for SG PX(Q) with TIE T, then (ps(f)(rc) = 0. This shows that 9z(f)W = (1-TC) JS Re(co(f)) G (;c-l)L(n,R) for all T I E P . Thus

(pz(f)

is a parabolic 1-cocycle and we obtain an

R-linear map

9 :5 Theorem 1 (Eichler-Shimura). Suppose that F is torsion-free. Then the map (p is a surjective isomorphism. Before going into the proof of this fact, we note an application. First of all, using the notation of Proposition A.I applied to the curve Y = T\H and its compactification X, for each commutative algebra R and for each R[F]-module M of finite type over R, we see that Ai(M) = HomR[r](Ai,M) is of finite type over R because Ai is free of finite rank over R[F]. Therefore H^(F,M) is of finite type over R. Here "*" means either the usual cohomology, the compactly supported one or the parabolic one. Note that we have exact sequences 0 -> Z^KJvl) -> Hom R[ r](Ai,M) -> Hom R[ r](A 2 ,M), Hom R[r ](Ao,M) -> B^KJVI) -> 0, and 0 -> B^K,]^) -* Hom R[r ](Ai,M). If A is an R-flat algebra (or module), tensoring by A the above sequences over R, we have and hence (la)

H 1 (r,M® R A)=H 1 (r,M)® R A if A is R-flat.

Here the sentence "A is R-flat" is a terminology in commutative algebra meaning that M® R A —» M'® R A —> M M ® R A is exact whenever M —> M' —> M" is exact as a sequence of R-modules ([Bourl, I]). By the exactness of o -> H^r.M) -> HHr.M) -> e s e S H 1 ^ , ] ^ ) (S = X-Y), we see that (lb) H1P(F,M)®RA = Hj,(r,M®RA) if A is R-flat. We record here a corollary of the theorem which is an easy consequence of (la,b).

6.2. Eichler-Shimura isomorphisms

169

Corollary 1. Let Lp (resp. Lj be the quotient of Hp(F,i:(n;Z)) (resp. H 1 (F,i:(n;Z))j by the maximal torsion subgroup of Hp(r,£(n;Z)) (resp, H^F^njZ))). Then the maximal torsion subgroup of Hp(F,i:(n;Z)) (resp. H1(F,i:(n;Z))>) is finite, and Lp (resp. Lj is isomorphic to the image of H^F^njZ)) (resp. H ^ i l f e Z ) ) ) in H^F^feR)) (resp. H ^ F ^ R ) ) ) . Moreover LP®ZR = Hp(F,£,(n;R)) and L®ZR =H1(F,i:(n;R)). Thus we can identify Lp with a Z-lattice of the R-vector -space 5k(F) for k = n+2 via (p. The injectivity of (p. Here we reproduce the proof given in [Sh, VIII]. We define a pairing on L(n;Z). Consider the symmetric or skew symmetric matrix e = (8n.i,j(-i)i(51))o
= l (x n , x n 4 y,..., yn). Regarding each entry un"V (xn'Y) of

where (resp.

) as a monomial of degree n of indeterminate u and v (resp.

x and y) and letting y e GL2(C) act on them as an element of L(n;C), we see that (*)

Y(un,un-1v,...,vn)0t{y(xn,xn-1y,...,yn)} = det(y)n ^ j

^j

We regard L(1;A) as the space of A-linear forms on the column vector space A2, i.e. the A-dual of A2. Now we have the pairing A 2 xA 2 given by [I I,I I] = detl

. We identify An+1 with the symmetric n-fold tensor

product (A2)®n and consider L(n;A) =L(l;A)(S>n to be the dual of (A 2 ) 0n . Identifying u n V with ei®(n-i)®e20i for the standard basis ex = ^1,0) and ^2 = \0,l), we have a pairing on (A2)®n whose matrix is given by 0. Since L(n;A) is the dual of (A 2 ) 0n whose basis dual to {X^Y1}! is given by {ei0(n"i)(8)e2<8)i}, we can define a pairing <,) : L(n;A)xL(n;A) -* A by (2a)

( X aiX^Y 1 ,^ i=0

j=0

k=0

/n\"^ as long as ( ^ ) e A. In particular, (2b)

<(X-zY)n,(X-zY)n> = X (-Dn"k(k)zn"kzk = (z-z)n. k=0

We see from (*) that for any y e

170

6: Modular forms and cohomology groups [pc,y] = [x,fy] for x , y e (A2)®"

and hence (2c)

(yP,Q) = ( P ^ Q ) for P, Q e L(n;A).

This induces a pairing via the cup product (see [Bd,IL7]) (3a)

< , ) : H1c(Y,i:(m;A))(8)H1(Y^(m;A)) -> H*(Y,A) s A.

The last isomorphism is the one in Corollary 1.1. The differential form realizing the class (Re(co(f)),Re(co(g))) can be made explicit as follows. Writing any polynomial P = Z^gaiX^Y 1 as a column vector Xao,...,^ which we denote by P, we know that (Re(co(f)),Re(co(g))) is represented by the closed form O(f,g) = tRe(co(f))A0-1Re(co(g)). Thus by Corollary 1.1, we have (Re(co(f)), Re(co(g))> = JYQ(f,g). Thus we get a pairing for f,g e 5n+2CO given by A(f,g) = J Y ^(f,g). We can compute A(f,g) in terms of the Petersson inner product: Note that co(f) = f(z)(X-zY)ndz. Thus, by (2b), we have, for the complex conjugation c, t

co(f)A0-1(cQ(g))c = -2V z l(fg)(z-z) n dxAdy = ( ^ V ^ f

(k = n+2). This shows that (3b) A(f,g) = -(-2V z T) n - 1 {(f,g)+(-l) n+1 (g,f)}, A(f,V : 4 n - 1 g) = 2nRe((f,g)) and A(f,V-l n g) = -2 n V = lIm((f,g)). In particular, the pairing A(f,g) on 5n+2(D is non-degenerate. We are going to show that if (p(f) = 0 in the cohomology group, then A(f,g) = 0 for all g e A+2(r). The injectivity of (p follows from this fact. We consider the following function for a constant vector a e L(n;R) (later we will specify the vector a suitably for our computation) and for a fixed point z e CK F(w) = fWRe(co(f)) + a. Then we see that F(y(w)) = JJ (w) Re(co(f))+a = J^)Re(co(f))+(Pz(f)(y)+a = JzWY*Re(co(f))+9z(f)(y)+a = pn(y)F(w)+(pz(f)(Y)+(l-y)a. Since z(f)(Y) = (Y-l)b. Thus by taking -b as a, we have F(y(w)) = yF(w) for all w. On the other hand, we see

6.2. Eichler-Shimura isomorphisms

171

that dF = co(f), where d is the exterior differential operator. Similarly, we define G(w) = JW Re(co(g)). Then we have dG = Re(co(g)) and Q(f,g) = 'dFAS-MG = dCF-e^dG) = dCF-G^ReCcotg))). Let Oo be the fundamental domain of Yo as in the proof of Proposition 1.1 (see (Tl-3) in Appendix). Then using the notation introduced there, we know that the boundary of Oo is of the form: 2®o = £

s e S

t. + YU

Kaj-Dsj + (Pj-Ds'j}.

Thus we have, shrinking the holes of Yo (i.e. Yo -> Y) A(f,g) = lim JLdC'F.ReCcoCg))) = lim Yo^X

Y0-»X

Since F(a(w)) = aF(w) for all a e F, we see for any 1-simplex A that = j A t a*F.0- 1 a*Re(co(g))

Thus we have

I

J;

= 0,

because F(w) is bounded near cusps and Re(co(g)) is rapidly decreasing at each cusp. This shows the vanishing of f and the injectivity of (p. By the dimension identity (6.1.4), we conclude that cp is also surjective. However this proof of surjectivity requires the Riemann-Roch theorem (the dimension formula (6.1.4)) which comes from algebraic geometry. We now give a sketch of a purely cohomological proof of the surjectivity, which is a version of the treatment given in [MM], [MSh] and [Ha] in our special case: The surjectivity of (p. We show the surjectivity of the scalar extension of (p to C: (4) 5 k ( r ) 0 5 k ( r ) c = Sk(T)®RC = ^ ^ Here 5k(F) = {f(z) |f e 5k(F)} with

denoting complex conjugation,

and Hp(Y,Z,(n;C)) is the natural image of the sheaf cohomology group H^(Y,.£(n;C)) of compact support in the usual sheaf cohomology group H^YjiXnjC)). We resort to the Hodge theory of Riemannian manifolds with boundary to prove the theorem. We refer for technical details to standard texts in differential geometry. The boundary exact sequence of Corollary A.2 combined with the isomorphism between the sheaf cohomology group H ^ Y ^ n j C ) ) and the group cohomology group H1(F,L(n;C)) (see Cor. A.I and Prop.A.4) tells us that Hp(Y,Z
172

6: Modular forms and cohomology groups

Hl(Y,L(n;C)) using the de Rham cohomology theory (i.e. H ^ Y ^ = Hj)R(Y,X(n;C)); see Theorem A.2 and Proposition A.4). We have already defined the differential form co(f) attached to f e 5k(F). For f e 5k(F)c, we define co(f) = f(z)(X-zY) n dz. Then co(f) for f e 5k(F) is holomorphic and for f e 5k(F)c is antiholomorphic. Anyway they are closed forms, which define cohomology classes in H^R (Y,£(n;C)). Thus we have a map

Identifying Hl(Y9L(n;C)) with H^FJLOnjC)) by the canonical isomorphism, it is not so difficult to show that Re(O(f)) = (p(f) for f e 5k(F) tracking down all the isomorphisms between various cohomology groups presented in Appendix. Thus by the first proof, O is injective. The point here is to prove the surjectivity of <& onto Hp(Y,£(n;C)). First we shall show that O takes values in Hp(Y,£(n;C)). If f e 5k(F)05k(F)c, we define for each cusp s e P^Q),

F s (z) = JsZco(f). This integral is well defined because f is decreasing exponentially towards the cusp s. As already seen, dFs = co(f). Note here that Fs(z) is not invariant under F; thus 7rlFs(y(z)) may be different from Fs(z). However F s does behave nicely under F s = {y e F | y(s) = s}, i.e. Y^FgCyCz)) = Fs(z) for y e F s . Thus F s is a section of L(n;C) on Us-{s} for a small neighborhood Us of s on X. Taking a C°° function (|)s such that its support is contained in Us and (|)s = 1 on a still smaller neighborhood of s, we define F = XseS^sFs. Then F is a smooth global section of £(n;C) and co(f)-dF is compactly supported. Thus the cohomology class of o(f) falls in Hp(Y,£(n;C)). Now we construct a Laplacian acting on the sheaf of smooth differential p-forms $ on Y with values in X(n;A) (for A = R and C). Since % : SL2(R) -> 0< given by X H X ( V - T ) induces an isomorphism TC : SL2(R)/SC>2(R) = 9{, we can consider a new covering space £'(n;C) = I\(SL2(R)xL(n;A))/SO2(R), where the action is given by y(x,P)u = (yxu,P | u) for (y,u) e FxSO2(R) with the right action of SL2(R) given by PI x(X,Y) = P((X,Y)lx). We claim: (5) We have an isomorphism of covering spaces of Y: £(n;A)/y= L'(n;A)/Y induced by the map: (x,P) H» (X,P | X) on SL2(R)xL(n;A) for A = R and C. Let us prove (5). The map is well defined because y(x,P) corresponds to (yx,yP I yx) = (yx,P) = (y,l)(x,P). It is obvious that the map induces an isomorphism at each fiber and hence, it is an isomorphism globally because both covering spaces are locally trivial.

6.2. Eichler-Shimura isomorphisms

173

Hereafter we identify L\n; A) with L(n; A) by the map (5). The merit of the new realization is that it is easy to give a canonical hermitian pairing to each fiber. Since SC>2(R) is a connected compact group, the image of SC>2(R) in GL(L(n;R)) is compact, which is thus contained in a compact orthogonal group of a positive definite symmetric form S on L(n;R). In fact, we can take as S the symmetric n-th tensor of the standard hermitian inner product: (P,Q) = P(ei) Q(e1)+P(e2) Q(e 2 ) on L(1;R), where ei = \lfl) and e 2 = l (0,l) make up the standard basis. Since S(P| U , Q | U) = S(P,Q) for u e SC>2(R), this product induces a positive definite hermitian product on each fiber of £'(n;C) (= L(n;C)). On 9{, we have as SL2(R)-invariant Riemannian metric y~2(dx2+dy2) = y"2dz<8>d z. Take any pair of C-valued differential forms {coi} 1=1,2 on a simply connected open set U in Y which gives an orthonormal basis at each fiber on U under the Riemannian metric. For example, coi = y-1dx and 0)2 = y~*dy are a good choice. Then we define *0>i = C0j ( i ^ j), *(COIAOL>2) = 1 and *1 = CO1ACO2. Extend this operator C-linearly to the space of differential forms defined on U. Then we know from de Rham that the operator "*" is independent of the choice of the basis (0)1,0)2) and hence extends to a global operator on the sheaf AQ1. Let us write F for £(n;C) and consider the sheaf J4$l of smooth differential forms with values in F. Since the sheaf £(n;C) is locally constant, the same procedure of defining the "*" operator works well for J%pl. For 0),r| e ^ F ! ( Y ) , writing *co I u = i(z)coi+(|)2(z)o)2 and r| = (pi(z)o)i+(p2(z)o)2, we define S(T],co) = XijS(<|)i,(pj)CQiA*G)j. Then S(r|,co) patches up together well to give a global 2-form. We define (r|,co) = J Y S(r|,0)). Then ( , ) is a positive hermitian form defined on the subspace of square integrable forms in . ^ ( Y ) . Similarly we can define a positive pairing ( , ) between .%°(Y) and -%2(Y). Let d1 (resp. d") be the holomorphic (resp. antiholomorphic) exterior differential operator. We then define the formal adjoint 5' and 8" of d1 and d" respectively. That is, (d'co,r|) = (CO,8'TI) (resp. (dM0),r|) = (co,8"r|)) for all square integrable co with square integrable d'o) (resp. d"co) and all square integrable r|. We put 8 = S'+S". Then 8 is the adjoint of the exterior differential operator d. We define the Laplacians A = d8+8d, A' = d'S'+S'd' and A" = d"8M+8"d". It is easy to see that A = A!+AM. We call 0) harmonic if Aco = O. Write co = S j f j (X-zY) n ' j (X-zY) j dz+I j f j (X-zY) n -J(X-zY) j dz. Then we have, for y e F, fj(Y(z)) = fj(z)j(Y,z)n-J+2j(y, z)J and fj(Y(z)) = fj(z)j(Y,z)n-Jj(Y,z)J+2, fa b^ where j(Y,z)=cz+d for y = I. If co is square integrable and harmonic, by the spectral theory of the unitary representation of SL2(R) on the L2-space L2(I\SL2(R)), it is known that each fj(z) and fj(z) and all their derivatives are exponentially decreasing as Im(oc(z)) -» ©o for all a e SL2(Z) [Ha]. The

174

6: Modular forms and cohomology groups

description of the spectral theory would take us beyond the scope of this book. Thus we just admit this fact and conclude the proof. By the exponential decay towards the cusps, if CO is square integrable and harmonic, it is easy to show A'co = A"co = 0. In particular, d'co = d"co = 0 and 5'co = 5Mco = 0 if co is harmonic and square integrable. Thus if co is harmonic and square integrable, co is a sum of holomorphic form and anti-holomorphic form. Thus we may assume that co is holomorphic (fj = 0). We now show that co = co(fo) for a holomorphic cusp form fo on T of weight n+2 if co is holomorphic. To do this, we compute 8' for co = Xjfj(X-zY)n'J(X-zY)Jdz. Here we have written down co as a section of L(n;C). The section of L'(n;C) corresponding to co is ( 1 = ^ = 1 , g € SL 2 (R)). Writing (|) = Z k (|) k (X-zY) n - k (X-zY) k for a C°°-global section of £(n;C), we have fk(y(z)) = fk(z)j(y,z)n-kj(Y,z)k and k=o

dz

Ziy

k=o Then the C°°-section of L\n;C) corresponding to d'cj> is given by k=o °z

Ziy

"W 2>(X-zY)n-kl(X-z k=0

We write 8n.k for the differential operator — + . Noting the fact: dz 2iy S((X-iY) n ~ j (X + iY) j ,(X-iY) n " k (X + iY)k) = 2 n 5 j>k and we have

It is easy to see that (see the proof of Theorem 10.1.2), for e = y 2 —, dz JYfk8n-k4>kyndxdy = J Y if k y n - 2 dxdy. Thus we have 5'co = £

(ef k +^f k+1 )(X-zY) n - k (X-zY) k

k=0

~ 3 k By a direct computation, we have for 8 k = -— dz 2iy d"co = j ; (5 k f k +^f k + 1 )(X-zY) n - k (X-zY) k , k=o

6.3. Hecke operators on cohomology groups

175

where we agree to assume that fn+i = 0. By the fact: d"co = 8'co = 0, we know that

f - -M - f - •*<•• This implies fi = 0 and fo is holomorphic. Again using d"co = 0, we conclude fj = 0 for j > 0. Thus co = co(fo) for foe 5k(F). If co is antiholomorphic, co = co(fo) for fo e 5k(F)c. Now we want to define Hodge operators. Let <|) be a square integrable smooth 1-form. Consider the L2-space L of 1-forms with values in L(n;C). Let *F be the L2-closure of (A(|) | Supp((|)) is compact} in L. Writing Tc(Aipl) for the space of compactly supported global sections for the sheaf .%*, we take, for any compactly supported co e j^p1» a unique form \\f e W such that I co-\|/ I = ^((o-\\r,(o-\\r) is minimal, i.e. \j/ is the orthogonal projection of CO in W. Then we put Hco = co-\}/. Then by definition (Hco,Ay) = 0 for all compactly supported y. This implies Hco is a harmonic current. Since A is an elliptic operator, any harmonic current is in fact an analytic function (cf. [DuS, §3] or [Ko, Appendix]). Then it is well known that we can find a smooth solution of the equation A|i = co-Hco because on *F the spectrum of A does not vanish. Hence A has a formal inverse defined on *F, which we write G [Ko, Appendix, §7]. Thus ji = Gco and HG = 0. That is, AGco = (l-H)co. Since d commutes with A, G commutes with d, and hence Hco is cohomologous to co if co is closed. As already shown, Hco is in the image of O. Thus O is surjective.

§6.3. Hecke operators on cohomology groups Let A = {a e M2(Z) | det(a) *0}. We can let the semi-group A act on H as follows. If det(oc) > 0, a acts on H by the usual linear fractional transfor(T mation. If a = j = , we put j(z) = -z. If det(a) < 0, we can decompose a = ajj with det(aj) > 0 and define oc(z) = (ocj)(j(z)). One can easily check that this action is well defined (i.e. associative). Actually, identifying tH with SL2(R)/SO2(R) = GL2(R)/O2(R), this action is the left multiplication by elements of GL 2 (R). Let a e A and ( r , F , a l ) be the semi-group in A generated by a 1 = (let(a)a"1 and two congruence subgroups F and F . For any (r,F,a l )-module M, we define the Hecke operator [FocF1] with det(oc)>0 acting on H^rjVI) as follows. First decompose FaF 1 = LIiFc^. For each y e T\ we can write a[j = yiOCj (yi^a* = ocjy1) for a unique j

176

6: Modular forms and cohomology groups

with Yi e F. Then for each inhomogeneous cocycle u : F -^ Ln(A) (see Appendix about cocycles), we define v = u I [FaF 1 ] by v(y) = Xi 0Cilu(Yi), where a 1 = det(a)a' 1 . For y,8 e F, we define yi and Si as above. Now let us check that v is a cocycle. Note that aiyS = YiSjOCk for some k. Thus v(y8) = liCCiVyiSj) = Ii{ai l yiu(5j)+ai l u(yi)} = Si(yr 1 a i ) l u(5j)+v(y) = I i (a j r 1 ) l u(5 j )+v(y) = yv(5)+v(y). This shows that v is a 1-cocycle of F'. If u is a 1-coboundary, i.e. u(y) = (y-l)x, then

v(y) = 2>il(Yi-l)x = Xj( a J r 1 ) l x - Xiailx = (y-l)]T ^ x and v is a 1-coboundary. This shows that the operator [FaF1] is a well defined linear operator on H ^ ^ M ) into H^F^M). Now if u is parabolic, by replacing y as above by any parabolic element % e P, we know from the above computation that if u(rc) = (7t-l)x, then V(TC) = (7C-l)Zi(Xilx. Thus v is again parabolic and [FaF1] sends Hp(F,M) into Hp(F',M). We define the multiplication in the abstract Hecke ring R(F,A) as in [Sh, 3.2] and [M, §2.7]. We see easily that [ F a F ] o [ r p r ] = [ ( F a r ) . ( r p r ) ] and hence R(F,A) acts on the cohomology group if M is a A-module. When F = F' = Fo(N) for a positive integer N and M is a A'l-module for A' = {T

h

]e

A | det(a) ?fc 0, c e N Z

and dZ+NZ = Z } ,

we define the Hecke operator T(n) for each integer n > 0 by the action of Z[FaF], where F a F runs over all double cosets in {a e A'11 det(a) = n}. For a fixed point

z e Oi, let us compute (pz(g)(Y) = J

Re(co(g))

for

g = f I [FaF] = Xif | ai (FaF = Ili FaO in terms of f e 5k(F). Note that 8 n (a(z)) = det(a)(cz+d)- n - 2 a5 n (z). Since scalar t of A acts on L(n;C) via the scalar multiplication by tn, we obtain a l Re(co(f))°cc = Re(co(f I a)), where f I a = det(a) k " 1 f(a(z))j(a,z)- k (k = n+2). From this fact, we know that

l F ( Z ) ai*Re(co(f)) = X i a i l H W Re(co(f)). As in the proof of Theorem 2.1, we put F(w) = JW Re(co(f))+a. Then the 1-cocycle u(y) = F(y(w))-yF(w) represents the cohomology class of cp(f). Then we see that

6.3.

Hecke operators on cohomology groups

177


for x = ZiOCilF((Xi(z)), since a ^ = yaj1. This shows that the isomorphism cp as in Theorem 2.1 is in fact an isomorphism of Hecke modules (i.e. compatible with Hecke operators). Similarly the isomorphism O : 5 k ( r ) 0 5 k ( r ) c = H}>(r,L(n;C)) is an isomorphism of Hecke modules. Let N be a positive integer and % : (Z/NZ)X -» A x be a character with values in a ring A. We define a new All-module L(n,%;A) as follows. We take L(n;A) as the underlying A-module of L(n,%;A) and define a new action of A'1 by, fa b\ writing the original action of y=\ € A1 on L(n;A) as y»P, c a v J f P = JC(d)yl.P. When we regard L(n,%;A) as a left ro(N)-module, the action is given by

YP = %(d)y.P for y = [* *) e r o (N). Theorem 1. For any positive integer N, we have natural isomorphisms of

Hecke modules O : 5 k (r o (N),x)e5 k (ro(N),x) c = H^roCNXUn^C)), where % denotes the complex conjugate of x and 5k(r o (N),x) c = { f ^ l f e 5 k ( r 0 ( N ) , x ) } . Proof. We can define the cocycle (pz(f) for f e 5k(ro(N),x)®5k(r o (N),x) c by

cpz(f)(Y) = JJ (Z) co(f). Then it is easy to check that (pz(f) in fact has values in X(n,x;C) (not just in £(n;C)). Thus we have a natural map associating the cohomology class of (pz(f) to f: H^roC^.LCn.XiC)). Now we take a small normal subgroup T of Fo(N) of finite index in Fi(N). We may assume that Y is torsion-free. Then % as a character of Fo(N) factors through the finite quotient G = r o (N)/T. The double coset n 2 r 0 ( N ) = r o (N) defines two trace operators:

Tr : 4(r,L(n;C)) -+ 4(ro(N),L(n,x;C)), Tr : 5 k (r)05 k (r) c -^ 5k(r0(N),%)e5k(r0(N),x)c.

178

6: Modular forms and cohomology groups

We have dropped the symbol % from the left-hand side of Tr, not only because L(n,%;C) = L(n;C) as F-module but also because the action of G given by the operator [FaF] = [Fa] for a e FQ(N) is defined relative to the action of a on L(n;C). Write "res" for the map given by restricting cocycles of FQ(N) to the smaller subgroup F. Then it is obvious that Tr°res is multiplication by the index [Fo(N):Fi(N)]. This shows that res induces an isomorphism from 4(T 0 (N),L(n,x;C)) (resp. 5k(Fo(N),x)®5k(Fo(N),x)c) onto H^(r > L(n;C))[x] = (x e H^FJLfeC)) | x | g = X(g)x for g e G} (resp. {5 k (F)05 k (F) c }[x] = {x G 5 k (F)05 k (F) c | x | g = %(g)x for g e G}). Thus we have a commutative diagram:

I inclusion c

O : {5 k (r)e5 k (r) }[%]

si res > H^TJLfo

Since the lower horizontal arrow is a surjective isomorphism of Hecke modules by Theorem 2.1, so is the upper top line.

Theorem

2. For every positive integer X

N

and every character

x

% : (Z/NZ) -» C , we have, if k > 2 5k(F0(N),x;A) = 5k(Fo(N),x;Z[x])®z[%]A and Hom Z [ X ](h k (Fo(N),x;A),A) = 5k(F0(N),x;A) by $ h^ ^

= 1

^)(T(n))qn

for any Z[%]-algebra A inside C or Q p . Proof. By Theorem 5.3.1, we have Hom c (h k (ro(N),x;C),C) = 5k(Fo(N),x) via $ h^ ]T~ =1 ^)(T(n))qn. On the other hand, by Theorem 1, hk(Fo(N),x;Z[x]) leaves stable the image L of H1p(T0(N),L(n,x;A)) in 5 k (Fo(N),x)e5 k (Fo(N),x) c . This implies h k (F 0 (N),x;C) = h k (F 0 (N),x;Z[x])®z [X ]C, and therefore Hom c (h k (Fo(N),x;C),C) = HomZ[X](hk(Fo(N),x;Z[x]),Z[x])®z[%]C. The image of HomZ[X](hk(Fo(N),x;Z[x]),Z[x]) in 5k(F0(N),x) is exactly the space 5k(Fo(N),x;Z[x]), and hence the assertion follows for A = C. We can deduce the assertion for general A from that for C in the same manner as in the proof of Corollary 5.4.1.

6.3. Hecke operators on cohomology groups

179

Now assume that N = p p a for a prime p. We now want to describe the similar result for M(x) = f^k(Fo(N),%). The map O is well defined even on ^k(Fo(N),%) and gives a commutative diagram for F = Fo(N) whose rows are exact: 0 -> S(T,x)®S(T,x)c lit O

^ M(x)®S(T,x)c 1<3>

-» Coker(i) I '

0 - » H 1 P (T,L(n,x;C)) - » H 1 (T,L(n,x;C)) -> e r e S H I 1 ( r . J L ( n f z ; C ) ) . We now claim, for any field K containing Q(x), that if a > 0 (l)H1(r0(ppa)s,L(n)x;K))=(K. [U,

if

« is equivalent to either otherwise.

or 0,

When s = 0 or °°, the assertion follows from the argument which proves (1.2a,b), since F s is generated by either 7ioo = L - or no =

v° v

modulo

a

VPP

U

the center {±12}. As seen in §1, S s ro(paq)\SL2(Z)/Foo = {ideals of Z/p a pZ} given by

The last isomorphism follows from the strong approximation theorem (Lemma 1.1) and the fact that the image of ro(pp a ) in SL2(Z/p a pZ) is the subgroup of upper triangular matrices which fixes one line in (Z/p a pZ) 2 . This implies that each cusp s which is equivalent to neither ©o nor 0 is represented by a u =

for uepZ-pp a Z. Then *. = a ^

^

=[ ^

l + ]±j

E r o (p a p)

for some h e Z. Since 7CS is a generator of ro(p a p) s , h is determined by the condition that u2h e p a p Z and | h | p is as large as possible. Thus we know that I h I p = max( | p a p I p I u | p " 2 ,l) > I p a p I p . Then by the primitivity of %, %(l+uh) * 1 and 7ts-l is invertible on L(n,%;K). This shows the vanishing of the cohomology because of H 1 (r 0 (p a p) s ,L(n,%;K)) =L(n,%;K)/(7Cs-l)L(n,%;K). If k > 2, we already know from Proposition 5.1.2 that the Fourier expansion of Ek(X) (resp. G(%)'1Gk(%)) has non-trivial constant term at 00 (resp. 0) and has no-constant term at 0 (resp. 00). We will see in Chapter 9 the same assertion for k = 2 provided that % * id. This shows that the map O' of Coker(i) to 0SEsH1(Fo(pa)s,L(n,%;R)) induced from O is in fact surjective. Comparing the dimension, O1 is an isomorphism. Thus we have

180

6: Modular forms and cohomology groups

T h e o r e m 3 . Let N = p p a with a > 0, k = n+2 and % be a primitive character modulo N. Then we have the following commutative diagram whose rows are exact: 0 -> 5 k ( x ) 0 5 k ( % ) c -^ 44(X)®5 k (%) c -> Coker(i) in O in O m

0 -* 4

^

H

Now we deal with the case of SL2(Z). When k > 2, the argument is completely the same as in the case of Theorem 3. When k = 2 (i.e. n = 0), the small circle around oo in X is bounded by Yo. Thus the natural restriction map: H1(SL2(Z),C) -»

ft\{±l}\l(Z)£)

[71 m^ I ] for U ( Z ) = j L I m e Z i s a zero map. This in particular means that the constant term of any modular form of weight 2 for SL2(Z) vanishes. Thus af 2 (SL 2 (Z)) = 5 2 (SL 2 (Z)) = {0}, and we have T h e o r e m 4. We have the following commutative diagram if k = n+2 > 2: 5 k (SL 2 (Z))05 k (SL 2 (Z)) c — ^ f * 4 ( S L 2 ( Z ) ) 0 5 k ( S L 2 ( Z ) ) c -> Coker(i) in O IU

HJ»(SL2(Z)JL(II;C))

> H 1 (SL 2 (Z),L(n;C)) -> HL(L(n;C)) -> 0,

where HL(L(n;C)) = H ^ f t l J U C Z X L f o C ) ) . WAen k = 2, we have the following commutative diagram: 5 2 (SL 2 (Z))052(SL 2 (Z)) C s ^ k ( S L 2 ( Z ) ) e 5 k ( S L 2 ( Z ) ) c Mi 4> iu O Hj,(SL 2 (Z),C) s

UHSL2(Z),C).

Then in the same manner as in the proof of Theorem 2, we have Corollary 1. For every prime power N = p a and every primitive character % : ( Z / N Z ) X -> C x , we have, if k > 2,

Let N > 4 be an integer. Then T = Fi(N) is torsion-free. We can consider for (I ( primes q the double coset action T(q) = [ F a F ] for a = Q on modular forms f on Fi(N), which is given by f | T(q) = ^ f | k a i for the decomposition F a F = U i F a i . In fact, we can choose oci so that Fo(N)aFo(N) = IIiFo(N)ai. Therefore we have two commutative diagrams:

6.3. Hecke operators on cohomology groups

181

ri(N))

4T(q) and H 1 (r 0 (N) > L(n,x;A)) c lT(q) 4T(q) 1 1 H (r0(N)>L(n)%;A)) c H (ri(N),L(n;A)). The finite group (Z/NZ)* = Ti(N)\ro(N)

acts on 314(1^ (N)) by

Similarly the operator (d) acts on the cohomology via the action of [F

. F].

Now we define T(n) for general positive integers n by a(m,f | T(n)) = X b | (m,n) ^-^(mn/b^f |
modNM2(Z)}

(see [Sh, III] or [M, §4.5]). Anyway we can define the Hecke algebra Hk(Fi(N);A) (resp. hkQTi(N);A)) for any subring A of C as an A-subalgebra of the A-torsion-free part of H^riCNJJLfaA)) (resp. HJ>(ri(N),L(n;A)) for n = k-2. Then, by the Eichler-Shimura isomorphism, these algebras act faithfully on f^k(Fi(N)) and 5k(Fi(N)). Note that we have not proven that fAfk(Fi(N);A) is stable under Hk(Fi(N);A) although it can be proven using a geometric interpretation of modular forms due to Katz [K5] (see also [HI, §1]). Anyway we have (2a) H k (Fi(N);A) = H k (r 1 (N);Z)® z A and h k (Fi(N);A) = h k (Fi(N);Z)® z A for k > 2. There are natural A-algebra homomorphisms (2b) Hk(Ti(N);A) -> Hk(Fo(N),%;A) and h k (Ti(N);A) -> hk(Fo(N),%;A), which take T(n) to T(n) and hence are surjective. These homomorphisms are obtained by restricting T(n) for Fi(N) to the space of modular forms for Fo(N). Now we want to describe the action of Hecke operators in terms of sheaf cohomology groups H£(Y,£(n,x;A)), HJ>(Y,Z,(n,x;A)) and H ^ Y ^ n x A ) ) . To include the most general case, we take a (F,F',a)l-module M for congruence sub-

182

6: Modular forms and cohomology groups

groups F and F and a e A. We consider the locally constant sheaf M associated with M. We can in fact split the operator [FaF'] on H^FjM) into three parts: [FaF 1 ] = Tr r/ d>ao[OaO a ]oresr/^, where O = a F ' a ^ n r ,
1

I. This can be checked as follows.

Note that Ff = UiO a 5i => a ^ F a F ' = Uia^FaSi => F a F ' = UiFa8i. Moreover if the first decomposition is disjoint, then the other two are also disjoint. Then by definition, it is clear that [FocF] = Trry<&a°[OaOa]°resrv. Exercise 1. Give a detailed proof of [FaF1] = Trryd>a°[®aOa]°resrv. We now construct the corresponding morphism of sheaves. The operator [<J>ocOa] is easy to take care. We write Y(O) for O W Then the map a* : 2/xM s (z,m) h-> ( a ' ^ z ) ^ ^ ) G 9{YM induces a morphism a* :

M/Y(<E>)

-> M/Y(#a) because

oc*(y(z,m)) = a^yao^fem). a

Thus we have [<X>ocO ] : H^(Y(O),M) -> H*(Y(Oa),M)- The restriction map is induced from the projection M/Y(O) ~-> M/Y(r> Now we define the trace operator. Since the projection % : Y(O a ) -> Y(F') is etale (i.e. it is a local homeomorphism), for each small open set U in a simply connected open set in Y(F), TC^CU) is a disjoint union of open sets each isomorphic to U. Thus taking an open subset Uo in 9-C isomorphically projected down to U, we know that rc*M(U) = M(Uo)^ r: ^ a ^ This isomorphism is given as follows. We may identify Tt'^U) with the image of the disjoint union Ui8i(Uo) for a disjoint decomposition F' = Ui<£a8i. Then we identify M(5i(U0)) = M with M(U 0 ) = M by M(8i(Uo)) 9 X H 8ilx e M(U 0 ) = M(U). Now it is clear that where Indrya]Z[F] with the F'-action given by y(m(8)a) = m^ay1. Since the direct image of a flabby sheaf is flabby by definition and TC* is an exact functor because n is a local homeomorphism, any flabby resolution of M/Y(<&«) gives rise to a flabby resolution of (7C*M)/Y(n J u s t by applying 71*. Thus we know that E£(Y(r),7C*M) = H^(Y(O a ),M). This gives in particular a proof of Shapiro's lemma in group cohomology asserting that for any pair of groups G D H and H-module M (3)

HkG.Indo/HCM)) = ff(H,M).

6.3. Hecke operators on cohomology groups

183

Now we define Tr : 7t*M/Y(r)(U) -» M(U)/y(r) by Tr(x) = Zi5ilx, which induces a morphism of sheaves. Obviously this is induced from the map tr : Z[r f ] -> Z [ O a ] : Tr = id®tr : Indrv^a(M) = M®Z[a]Z[r] -> M. Then Tr induces a map of cohomology groups:

Tr : t

^

i

We define [ T a r ] : H^YCD.M) -> H i (Y(r f ),M) by Trryao[OaOa]oresr/. It is tautological to check that this action of the Hecke ring is compatible with the canonical isomorphisms between sheaf cohomology and group cohomology. Let us now assume that M is also an A-module for a commutative ring A. Then, writing M* for the A-dual of M, we have a pairing by the cup product < , >r : H^(Y,M)®H1(Y,M*) -> H*(Y,A) = A

(Y = Y0T) = T\H).

First we suppose that A is a Q-algebra. Then this pairing is non-degenerate (for example, extending scalars to C and then it is clear for M = L(n,v;C) = M* (see (2.3a)); see [Bd,IL7] for a proof in general). We have a commutative diagram for a subgroup O of F, Indr/d>(M®AM*) ii

\i < , >

< , ) : Ind r /^(M)® A M* -> A[O\T] i Tr(8>id 4 Tr ( , ) : M®AM*

> A.

Note that Tr : A[O\T] —» A induces an identity on A = H ^ Y ^ J . A ) = HjCY^.AtOVn) ~> Hj(Y(D,A) = A. This is an easy consequence of Proposition 1.1 and its proof. Then the above diagram induces another commutative diagram:

Tid®res

"

< , ) : H ^ Y ^ M ) ® ^ " ^ ^ ) , ^ ) -> A

( , ) : H^YCn^DiSH^Wn^Ii) -» A. We thus have (4a)

(Trr/
184

6: Modular forms and cohomology groups

We see easily that (4b)

(x | [OaOa],y)(Da = <x,y | [Oaa*O])
In particular, when M = L(n,%;A), we identify M* with the dual lattice under <,) in L(n,%~1;A®Q). From (2.2c), (ax,y> = <x,aly>, we see that (4c)

(x | [OaOa],y)d>a = (x,y | [O a a l O])o for O a = a ' ^ a .

This implies, for x e HJ.(Y(r),M) and y G H ^ Y t r X M i ) , (4d)

(x | [r<xr],y> r = <x,y I [r<x l r]> r .

Now we specialize our argument to the case where M = L(n,%;A). We write L*(n,%-1;A) for the dual module in L(n,%"1;A<8>Q) under ( , ) . It is easy to see from (2.2a) that

(5)

L^n.x^A) = XLo A (i) X n " i Y i

in

L^'X"1^)-

Since r o ( N ) a l r o ( N ) = Tr o (N)ar o (N)x- 1 for % = ( ° ~*\ if we modify the pairing ( , ) and define a new one ( , ) by (6a) we have (6b)

(x,y) = (x | T,y), (x | T(n),y) = (x,y | T(n)).

Since ( , ) is non-degenerate for any field of characteristic 0, we see from Theorem 5.3.2 and the Eichler-Shimura isomorphism Theorem 5. Suppose that either % is primitive modulo p a or a = 0. Then H^(Yo(pa),i:(n,x;K)), H1P(Yo(pa),i:(n,%;K)) and H1(Y0(pa),An,%;K)) are all semi-simple Hecke modules provided that K is a field of characteristic 0. Proof. Let K/F be a field extension of characteristic 0. Then the assertion for K is equivalent to that for F. If the result is known for K = Q, then the assertion is true for all fields of characteristic 0 by extending scalars to K. To prove the assertion for K = Q, extending scalars to C, we may assume that K = C. By Theorem 5.3.2 and the Eichler-Shimura isomorphism (Theorems 1-4), we know that the assertion for Hp and H1. Then the assertion for the compact supported cohomology group follows from the duality (6b) compatible with the Hecke module structure on H1.

6.3. Hecke operators on cohomology groups

185

We continue to study the Hecke module structure of cohomology groups. We know that the following isomorphisms of Hecke modules: *4(r o (p a ),X;K) s Hom K (H k (r 0 (p a ),x;K),K), 5 k (r 0 (p a ),x;K) = Hom K (h k (r 0 (p a ),x;K),K). Since 5 k (r 0 (p a ),x;K c ) c = 5 k (r 0 (p a ),x;K) as Hecke modules via f <-> f, Hl?(To(va)Mn,X;K))

s Hom K (h k (r 0 (p a ),x;K),K) 2 .

Since hk(ro(pa),%;K) is semi-simple, we know that (7)

Hom K (h k (r 0 (p a ),x;K),K) = h k (r 0 (p a ),%;K) as Hecke modules,

because any semi-simple algebra S over K is self-dual under the pairing (x,y) = Tr K /Q(xy). Thus HJ ) (r 0 (p a ),L(n,%;K)) is free of rank two over h k (r 0 (p a ),x;K). Thus we have Corollary 2. Suppose that K is a field of characteristic 0 and % is a primitive character modulo p a . Then Hp(ro(pa),L(n,%;K)) is free of rank two over the Hecke algebra hk(ro(pa),%;K). Moreover, we have the following isomorphisms as Hecke modules: if either % ^ id or k > 2

H1(r0(pa),L(n,z;K)) = ^(ToCp^LfottK)) = hk(r0(pa),x;K)eHk(r0(pa),x;K) and if % = id and k = 2 ^ X K )

= H^(SL 2 (Z),K) s h k ( r 0 ( p a ) , x ; K ) 2 = {0}.

Since f^ k (r 0 (p a ),x;K) = Hom K (H k (r 0 (p a ),x;K),K) s H k (r 0 (p a ),x;K) and 5 k (r 0 (p a ),x) = Hom K (h k (r 0 (p a ),x;K),K) = h k (r 0 (p a ),x;K) as Hecke module and since these Hecke algebras are all semi-simple, we see that

(8)

Hk(r0(Pa),x;K) = hk(r0(pa),x;K)eEk(r0(pa),x;K)

for an algebra direct summand Ek(To(pa),x;K). Let E be the idempotent of Ek(Fo(pa),x;K) in Hk(ro(pa),x;K). To get each of the following exact sequences in one line, we drop FoCp06) from the notation of the following cohomology groups in (9b,c) and (10) and also denote by Hg1 for the i-th cohomology group for ro(p a ) s (s G S). Then the exact sequences of Hecke modules (9a) (9b) (9c)

0 -> 5 k (r 0 (p a ),x;K) \ f^ k (r 0 (p a ),x;K) -> Coker(i) -> 0, 0 -> Hj>(L(n,x;K)) -> H^Un.zsK)) -> e s ^ H s H U n ^ j K ) ) -^ 0, 0 -> 0 S G S H s o (L(n,x;K)) ^ H^(L(n,x;K)) ^ H ^ L f r j c * ) ) -> 0,

186

6: Modular forms and cohomology groups

are all split as Hecke modules by the idempotent E unless % = id and k = 2 for the last two sequences. In the special case of % = id and k = 2, we have (9d)

H 1 (SL 2 (Z),K) s H*(SL 2 (Z),K) = H ^ S L ^ Z ^ K ) .

Now we consider the action of j =

0

l,

on Hp(ro(pa),L(n,%,K)). Since j

normalizes r o (p a ), j acts on Hp(ro(pa),L(n,%,K)) via [r o (p a )jr o (p a )]. Then j 2 = 1. Since j{a e Atl | det(a) = njj" 1 = {a e A'11 det(a) = n}, j commutes with T(n). Thus the eigenspaces of j (10)

H^LOi.x.K))* = {x e H1P(L(n>x,K)) | x | j = ±(-l) n+1 x}

is naturally a Hecke module. When K = C, we see that the action of j is given by co — f > jl(j*co) at the level of differential forms. This is just interpreted in terms of modular forms as f(z) h-» f(-z). Thus j brings holomorphic modular forms onto anti-holomorphic ones. Then the Krull-Schmidt theorem tells us that H^(r o (p a ),L(n,x,C)) ± is free of rank one over h k (r 0 (p a ),%;C). By the semi-simplicity of hk(ro(pa),%;K), the same assertion is true for all fields K of characteristic 0. Thus we have, for all fields K of characteristic 0, that (11)

H1P(ro(pa),L(n,%,K))± is free of rank one over h k(r 0(p a ),%;K).

§6.4. Algebraicity theorem for standard L-functions of GL(2) In this section, we prove the algebraicity result for the Mellin transform of holomorphic modular forms which are called the standard L-functions of GL(2). In the following section, we construct the p-adic standard L-function of GL(2) attached to modular forms of weight k > 2. Thus, in the rest of this chapter, we fix a prime p and embeddings Q —» C and Q -> Q p . For simplicity, we only deal with modular forms in 5k(ro(pa),%) for a primitive character % modulo p a . This restriction is caused by our neglecting to cover the theory of primitive (or new) forms of arbitrary level N. Since such theory is fully expounded in [M], it is strongly recommended to the reader to carry out our construction for primitive forms of arbitrary level (using the theory in [M]). Here we understand that % = id and Fo(p0) = SL2(Z) when a = 0. As seen in Theorem 5.3.2, hk(ro(pa),%;C) is semi-simple, and 5k(ro(pa),%) is spanned by common eigenforms of all Hecke operators T(n). Let f be one of these common eigenforms. We also know that, if we take the Q(%)-algebra homomorphism X : h k (r 0 (p a ),%;Q(%)) -> C given by f I T(n) = A,(T(n))f, then f

6.4. Algebraicity theorem for standard L-functions of GL(2) is a constant multiple of IT

187

A,(T(n))qn. The form f with a(n,f) = A,(T(n)) is

n=l

called a normalized eigenform. We fix such a X and its normalized eigenform f. We write Q(X) (resp. Qp(?0) for the subfield of Q (resp. Q p ) generated by ^(T(n)) for all n over Q (resp. Q p ). Let O be the p-adic integer ring of QP(X) and put V= OTIQ(^). By Theorem 3.2, the field Q(X) (resp. QP(X)) is a finite extension of Q (resp. Q p ). We now consider a new compactification X* of Y = ro(p a )V# Since F S W is a cylinder isomorphic to the space T in §4.1, we add to Y a circle S 1 at each cusp S E S as we did in §4.1 for T at /<*> and write this compactification X*. This type of compactification is called the Borel-Serre compactification of Y. Then for each r e Q, the vertical line cr connecting r and the cusp °° is a relative cycle in H 1 (X*,3X*;Z), where 3X* = UsesS 1 . Identifying cr with R+ = {x e R | x > 0}, we then have a natural morphism induced from R+ = cr —> X, (1)

Int r : H* (Y,£(n,x;A)) -> H* (R+,£(n,%;A)) = L(n;A).

This morphism is realized by the integration co h-» Jcrco for closed forms co. When ro(p a ) has non-trivial torsion, we just take a normal subgroup T of finite index of ri(p a ) and we define H* (Y,£(n,%;A)) to be the image of the restriction map in We have a natural surjection n : H*(Y,£(n,%;A)) —> Hp(Y,£(n,%;A)). We want to show that there exists a section (defined over K = A<8>zQ) i : Hp(Y,j£(n,x;A)) -» Hlc(Y ,L(nf%;K)) which is compatible with Hecke operators. We already know from (3.8) that n has a unique section of Hecke modules if A is a Q-algebra. Now we let A be a subalgebra of Q or Q p containing the integer ring of Q(X) and write K for the quotient field of A. Thus writing E for the idempotent of the Eisenstein part Ek(ro(pa),%;A) in Hk(ro(pa)OC;A), we can find 0 * r | E A such that K]E e H k (r o (p a ),%;A). Since the splitting of K over K is given by this E (3.8c), we know that i = 1-E: Hp(Y,X(n,z;A))-> Hc(Y,X(n,%;K)) satisfies 7ioi = id, and r|oi has values in H^(Y,£(n,%;A)). Now suppose that A is a principal ideal domain. We put H1p(Y,£
188

6: Modular forms and cohomology groups * (A,) e C x as follows. For f = X~_ MT(n))q n e 5 k (r 0 (p a ),x),

define co±(A) = Then 0 *


co(f)±(-l)n+1co(f)lj 2 •

H ^ Y ^ n ^ C ) ) ^ ] , and we define Qr(X) = Q*(k) by ±

(2)

For each field automorphism a of C, we always agree to choose 8+(A,°) to be 5±(Xf, where Xc : h k (r 0 (p a ),% a ;Q(%)) -> C is the Q(%a)-algebra homomorphism given by ?to(T(n)) = A,(T(n))° for all n. Anyway the period Qr(k) is determined up to units in A. We just fix one so that the above compatibility condition holds for the Galois action. Since co+(A,) is exponentially decreasing at each cusp, we know that (3a)

Intr(i(G>±(A.))) = Jcrco±(X) = Q ± a)Int r (i(8(^))) e Q±(?i)L(n;K).

Note that (3b)

^(co^)))

n

Exercise 1. Give a detailed proof of (3a). Now we compute the value (3b). Note that (O±(k) = 2"1(27C-V:^r)(f(z)(X-zY)ndz±f(-z)(X-zY)ndz). This shows that Into(co±(?i)) = 2-1iJ

This shows in particular that (-j—•

:—^—e TJ^A for j = 0,1,..., n, where

the sign of Q^iX) is given by the sign of (-iy. Let \}/ be any primitive Dirichlet character modulo N. Then a computation similar to (4.1.6c) (see also Corollary 5.5.1) shows

This shows

6.5. Mazur's p-adic Mellin transforms

189

Theorem 1. Let X : h k (Fo(p a ),x;Z[%]) -> C be a Z[%]-algebra homomorphism. Suppose that either a = 0 or % is primitive. Then for each Dirichlet character \\f and each integer j with 0 < j < n (n = k-2J, we have

and for every a e Gal(Q/Q), where the sign of QHX) is given by the sign of (-l)V(-l). Moreover, the p-adic absolute value I S(j,X<S>\|/) I p is bounded independently of \j/ and j .

§6.5. Mazur's p-adic Mellin transforms We are now ready to construct p-adic standard L-functions. Such L-functions were first constructed by Mazur for weight 2 forms in [Mzl] and [MzS]. It was then generalized to higher weight modular forms by Manin [Mnl,2]. For further study and conjectures concerning the materials here, we refer to the paper of Mazur, Tate and Teitelbaum [MTT]. We give an exposition of the construction using the method of modular symbols following the formulation in [Ki], which is quite similar to the one we have already given for abelian L-functions in §4.4. We shall use the same notation introduced in the previous section. In particular, X : hk(To(pa),%;Z[%]) —» Q is a Z[%]-algebra homomorphism for a primitive character % modulo p a (we assume that % = id if a = 0). Let O be the p-adic integer ring of QP(A,). We put K = Q(X) and A = OflK. Then A is a discrete valuation ring and Q T ( ^ ) is well defined. We assume the following ordinarity condition necessary to have a good p-adic L-function of X: (Ordp)

U(T(p))|p=l.

The algebra homomorphism satisfying this condition is called "ordinary" or "p-ordinary". We can construct a standard "p-adic L-function" without assuming (Ordp) (see [MTT]). However, the function obtained is not an Iwasawa function (i.e. is not of the form O(us-1) for a power series O e 0[[T]]). For the moment, we assume that a > 0. Recalling that cr is the relative cycle represented by the vertical line from r e Q to /«> on the Borel-Serre compactification X* of Y = Fo(pa)V^ we consider the map (1)

c : p - Z = U r = i P " l z -> HomK(H1c(Y,i:(n,x;K)),L(n;K)) given by c(r)(co) = Intr(co).

190

6: Modular forms and cohomology groups

We define for each COG H*(Y,£(n,x;K)), c© : p'°°Z -> L(n;K) by

Ca>(r)=(J 7) c ( r ) ( C 0 ) Then c
(2)

* (B (z+p m Zp) = a p - m ^ o

j)c«(-ij) for z = l , 2 , . . . prime to p.

This is well defined because c(r+l) = c(r). If we write G = Z p x and fix an isomorphism G = |ixZ p with a finite group JJ., we see that M- = (C ^

ZpXlC^^l}

where 9 is the Euler function and p = 4 or p according as p = 2 or not. Then the subgroup G a = l + p a p Z p corresponds to p a Z p . Thus (2) is tantamount to giving the value of the distribution O© on the standard fundamental system of open sets. To show that
= [Q JXJc(x/p)(co | T(p)) = apCco(x) and

XjTi1 ^«(x+jp m +p m+1 Z p ) = O tt (x+p m Z). This shows the necessity of assuming I ap | p = 1 to have a measure, not just a distribution. By a similar argument, we see that

(3) |0) t 0 (z+p m Z p )| p = |ap-mfpom j W l ) | p = | r™ ~ZJlntz/pm(co)|p is bounded independent of z and m. Thus <J>
6.5. Mazur's p-adic Mellin transforms

191

of Oo, we get a measure (pcoj. Now we want to show dO(o,j(x) = xMcp^.o- To show this, we may assume that co is integral (i.e. co has coefficients in L(n,%; A)) by multiplying co by a constant if necessary). We follow the argument given in [Ki] which originates from Manin [Mnl,2], For each <>| e (T(Zpx;A), take a locally constant function ^ : (Z/p n(k) Z) x -> A such that |fa~tyI p < p"k and n(k) > k. Then we know that

I P < I O
£<M)-n(k) z=l,(p,z)=l p n ( k ) -l

P

/ n(k)

z=l, (p,z)=l m p n ( k ) -l j=0 Z=l,(p,2)=l

mod p k z=l, (p,z)=l m

^ X J(t>(z)zjd(Pa),o(z)(Ij1)xm-JYJ mod p k , j=o

where Cj is the coefficient of c in Xn"JYJ. Thus taking the limit making k —> oo, we see that

(4)

J W c o j = J(|)(z)zjd(pG),o(z) for all $ e C(Z p x ;K).

Now we take co = i(8±(^)). Then we write O© as O 1 = O~ and compute the integral jcjxiO1 for each primitive character ty of (Z/prZ)x. We see that

(5)

192

6: Modular forms and cohomology groups

Thus projecting down to the coefficient in ( • jXm

of -O*, we get by (4) a

measure cp* = cp~ satisfying, for all characters <|): (Z/p r Z) x —» Kx,

(6)

J^zfd^z) =MT(p))VjG(^)^);^"1)

if 0 < j < k-1

and <|)(-l)(-iy has the same sign as that of (p*, and J(l)(z)zjd(p±(z) = 0 if the sign of <|>(-l)(-iy does not match. Now we suppose that a = 0. By the ordinarity assumption, we know that one of the roots of X -^(T(p))X+pk"1 = 0, say a, is a p-adic unit and the other one is non-unit, because k > 2. We write b for the other root. Then we define f (z) = f(z)-bf(pz) for f = S~ k(T(n))qn. Then f e 5 k (r 0 (p)). It is easy to n=l

verify that f | T(p) = af and f | T(n) = k(T(n))f for all n prime to p. Exercise 1. Give a detailed proof of the above fact. (Note the T(p) of level 1 and T(p) of level p are different.) We now use co = 8f±(A,) = ^ ± (^)' 1 co(f) to construct a measure. By the same computation, we get for each integer j with 0 < j < k-1

(7)

J
W 1

^

^

,

^

}

if the

«*»

of

cj>(-l)(-iy is equal to the sign of (p1, and J(|)(z)zM(p±(z) = 0 if the sign of (IK-lX-iy does not match the sign of (p1. Summing up all these discussions, we get Theorem 1. Let p be a prime, and X : hk(ro(pa),%;Z[%]) -> Q be a Z[%]algebra homomorphism for a primitive character % modulo p a (we a/fow that % = id if a = 0). Then we have two p-adic measures (f>\± on Z p x satisfying the evaluation formulas in (6) and (7). We now define the p-adic L-function for X and a primitive character \j/ modulo pP as follows. We take the transcendental factor either £2A + (^) or Q A (X) according to the sign of \|/(-l). We also take the measure either
= Lxv^zXzrMcpv.

6.5. Mazur's p-adic Mellin transforms

193

Corollary 1. Let the notation be as above. In particular, let \|/ be a primitive character modulo p p . Let X : h k (r 0 (p a ),%;Z[%])-> Qfor k > 2 be a Z[%]algebra homomorphism for a primitive character % modulo p a . Then we have an evaluation formula: if either a > 0 or $>0,then

a = (3 = 0, then

where a is the unique p-adic unit root of the equation X For further study of this type of p-adic L-functions, see [Ki] and [GS].

Chapter 7. Ordinary A-adic forms, two variable p-adic Rankin products and Galois representations A typical problem of p-adic number theory is the problem of p-adic interpolation, which can be stated as follows: For a given complex analytic function f(s) whose values at infinitely many integer points k are algebraic numbers, is there some p-adically convergent power series F(s) (with coefficients in a p-adic field) of p-adic variable s such that F(k) = f(k) for all integers k such that f(k) is algebraic? Many successful answers to this problem have already been discussed in Chapters 3, 4 and 6. Instead of taking complex analytic functions, we take complex analytic modular forms here and consider the problem of p-adic interpolation. We present, in this chapter and Chapter 10, some recent developments in the theory of p-adic modular forms, in particular, (i) p-adic analytic parametrization of classical modular forms, (ii) p-adic L-functions attached to each p-adically parametrized family of modular forms and (iii) Galois representations of these families of modular forms. Let us briefly explain what the p-adic family of modular forms is. As already studied in Chapter 2, a typical example of modular forms is given by the absolutely convergent Eisenstein series (see Chapter 5):

E k (z) = 2- 1 C(l-k) + Xr=1CTk-i(n)qn (k> 2), where crm(n) = X 0 < d | n ^ m *s ^ e s u m °^ m ~ t n powers of divisors of n. We modify tfk-i(n) by removing the (k-l)-th powers of the divisors d divisible by p. Then the modified coefficient G^Vifa) = Zo o^pVi(n) as the restriction of a continuous function o^ s .i(n) of the variable s which varies in the p-adic integer ring Z p . Actually, this dependence on the weight k is p-adic analytic; that is, the continuous function which induces s h-» a(p^s_i(n) can be expanded at each s e Z p into a power series p-adically convergent on a neighborhood of s. Then the formal Fourier expansion

may be considered as a solution to the problem of p-adic interpolation for the Eisenstein series Ek, where £p(s) is the p-adic Riemann zeta function given in Theorem 3.5.2. Note that E(k) = Ek(z)-pk"1Ek(pz) because of the modification to a(p)k_i(n) from ok_i(n).

7.1. p-adic families of Eisenstein series

195

For the moment, let us define naively a p-adic family of modular forms {fk} to be an infinite set of modular forms parametrized by the weight k whose Fourier coefficients depend p-adic analytically on the weight k. Later, we shall give a more precise definition. In fact, in the case of {Ek}, the coefficients are integers and thus can be considered as complex numbers as well as p-adic numbers automatically. In the general case, the coefficients of fk may not be just integers, and in fact there are many examples of such families with algebraic Fourier coefficients. With this general case in mind, we fix, once and for all, an algebraic closure Q p of the p-adic field Q p and an embedding of Q into Q p . Thus we can discuss the p-adic analyticity of Fourier coefficients of fk relative to k. To each modular form f = XI=o an w e a

nave

associated in Chapter 5 an

s

L-function L(s,f) = XI=i nfl~ , and for each pair of modular forms f and g = SI=o b n q n , we have another L-function Z)(s,f,g) = XI=i ant>nn~s- Once such a p-adic family of modular forms {fk} is given, it is natural to ask the problem of p-adic interpolation of the values (or more precisely their algebraic part) of {L(m,fk)} and {D(m,fk,f/)} by varying the weight k. We shall treat this problem for D(s,f,g) later in this chapter and in Chapter 10 (as for the treatment for L(m,fk), see [Ki] and [GS]). As for the p-adic interpolation of Galois representations, often one can canonically attach a Galois representation n^ (of Gal(Q/Q)) into GL2(QP) to each element fk in the family. Then we may also consider the problem of p-adic interpolation of the function k h^ 7Ck(a) e Mfe( Q p ) for any fixed a e Gal(Q/Q). If one succeeds in interpolating the function k h-> Ttk(tf) for every a, one may eventually obtain a big Galois representation into the matrix ring over the ring of analytic functions on Z p . We shall formulate this problem more clearly later, in §7.5.

§7.1. p-Adic families of Eisenstein series Here, we study the p-adic family of modular forms given by Eisenstein series. We write F for Fo(N) or Fi(N). Since we have already fixed embeddings of Q into Q p and C, any algebraic number in Q can be regarded as a complex number as well as a p-adic number. We fix a base ring O, which is the p-adic integer ring of a finite extension of Q p . Sometimes we need to consider the completion £1 of Q p under | | p (which is known to be algebraically closed). We fix a character \|/ = coa of ( Z / p Z ) x (p = 4 when p = 2 and p = p otherwise) for the Teichmiiller character co. We mean by a p-adic analytic family (of character \\f) an infinite set of modular forms {fk}°° k=M

teger M satisfying the following three conditions:

for some positive in-

196 (Al) (A2) (A3)

7: A-adic forms, Rankin products and Galois representations fk a(n,f k )e Q for all n, there exists a power series A(n;X) e O[[X]] for each n > 0 such that a(n,fk) = A(n;uk-1) for all k > M,

where u = 1+p (which is a topological generator of the multiplicative group W = l + p Z p ) . The family {fk} is called cuspidal if fk is a cusp form for almost all k (i.e. except finitely many positive k). Note that u k -l = (l+p) k -l is always divisible by p, and hence |u k -l l p < 1. The convergence of A(n;uk-1) follows from this fact. We now introduce the space of p-adic modular forms. First, we already know (see Theorem 5.2.1, Corollary 5.4.1, Theorem 6.3.2, Corollary 6.3.1) that if k > 2 (1)

*4(r o (pp a ),x) = ^k(r o (pp a ),%;Z[x])®z[ X ]C and

The assertion (1) holds for any subring A of C containing Z[%]. Thus we can define the space fWk(ro(ppa),%;A) for any ring A (inside Q or Q p ) by the right-hand side of (1). Since ^4(ro(ppa),%;Z[%]) is naturally embedded into the power series ring Z[%][[q]] via q-expansion, we can regard #4(ro(ppa),%;A) as a subspace of the power series ring A[[q]]. For each f e fA4(ro(ppa),%;A), its q-expansion will be written as f(q) = Xr=o a(n,f)qn. Let A be the one variable power series ring O[[X]] with coefficients in O. We call a formal q-expansion F(q) = Z°° A(n,F;X)qn e A[[q]] a A-adic form of character \j/ if the following condition is satisfied: (A)

the formal q-expansion F(uk-1) gives the q-expansion of a modular form in f^4(ro(q),\|/co'k;0) for all but finitely many positive integers k.

This is the definition of A-adic forms given in [Wil]. A A-adic form F is called a A-adic cusp form if F(uk-1) is a cusp form for almost all k. We will see later that there exists a A-adic cusp form which specializes to a non-cuspidal form at k = 1. By our definition, to give a p-adic family of modular forms {fk} is to give a simultaneous p-adic interpolation of their Fourier coefficients by the power series A(n;X). That is, by evaluating the p-adic analytic functions A(n;us-1) at integers k > a, we get the n-th Fourier coefficient of the modular form fk. When we defined a p-adic analytic family {fk}, we required an extra condition that fk = F(uk-1) is a classical (complex analytic) modular form for almost all k.

7.1. p-adic families of Eisenstein series

197

Thus a A-adic form F gives rise to a p-adic family {F(uk-1)} if F(uk-1) is a classical form for all but finitely many k. Now we want to construct an example of a p-adic family out of the set of Eisenstein series {Ek}. Since the n-th coefficient of Ek for positive n is a sum of (k-l)-th powers of divisors of n, we first show, for a positive integer a prime to p, the existence of a power series O(X) such that O(u k -l) = ak (u = 1+p) for integers k. We consider the binomial power series:

As seen in Chapter 3, (1+X)S is a power series with coefficients in Z p . This power series converges in the interior of the unit disk. Thus we can define the p-adic power 7s = (l+(y-l)) s (for y € l+pZ p = W) with exponent s e Z p and a morphism from the additive group Z p into the group of one-units W = l + p Z p by () As seen in §1.3, the p-adic logarithm function log induces an isomorphism W = pZ p . As a power series, we have an identity log((l+X)s) = slog(l+X); thus, also as a map, log(us) = sZog(u). Note that I log(z) I p < I p I p for any z G W and I log(\x) I p = I p I p * 0. Therefore, for any given z e W, by putting s(z) = log(z)/log(u), we have s(z) e Z p . Thus s : W = Z p . Therefore we can write z = u s(z) = (l+(u-l)) s(z) . Thus if an integer d satisfies the congruence d = 1 mod p, then we can write d = u s(d) , and for the power series A d (X) = d we have Ad(uk-1) = d"V ( d ) k = dkA. Therefore Ad(X) has the desired property for d when d = 1 mod p. To treat the case of d not congruent to 1 mod p, we use the decomposition Z p x = WX|LL introduced in §3.5 and the associated projection X H ( X ) = © ( X ) " ^ of Z p x to W. For each integer d prime to p, we put

Then we know that Ad(uk-1) = d V ^ * = d-1
Av(n;X) = £ 0 < d , n

(pd)=1 V (d)A d (X).

198

7: A-adic forms, Rankin products and Galois representations

This power series Av(n;X) is quite near to the power series we wanted to find, since Ay(n;u k -1) = G^ k -i(n) if k = a m o d ( p ( p ) for the Euler function (p. More generally, we have A ¥ (n;u k -1) = Xo 0. Now we consider the p-adic interpolation of a constant term. We consider the Dirichlet L-function As seen in §3.6, we have that (2a)

There exists a power series OV(X) in Zp[[X]] for each character \j/ of (Z/pZ) x with \|/(-l) = 1 such that for all integer k > 1 k |(l-\|/co- k (p)p k - 1 )L(l-k, V co- k ) if V * id, V k k k k 1 k " t<E>id(u -l) = (u -l)(l-co- (p)p " )L(l-k,co- ) if \j/ = id.

Let A = O[[X]] and define the A-adic Eisenstein series E(\|/)(X) e A[[q]] for each even character \\f = coa with 0 < a < p-1 by

where A v (0;X) = O ¥ (X)/2 if y = coa * id and Aid(0;X) = O id (X)/2X. Proposition 1. For each positive even integer k > 2 with k s a mod 9(p), we have E(\|/)(u k -l) = E k (z)-p k - 1 E k (pz)=E k (l p )€ fMk(ro(p)) in Q[[q]], where i p w f/ze trivial character modulo p. M<9r^ generally, for each non-trivial Dirichlet character % : (Z/papZ)x -^ Q x w/f/z % \ ^ = \\f and for k > 1, we have E(\i/)(%(u)uk-l) = Ek(%co"k) e f^k(r0(pap),%co-k). This proposition shows that we get the classical Eisenstein series even from the specialization at %(u)uk-l with %(u) ^ 1, which is not included in the definition of p-adic analytic families and A-adic forms. When \|/ = id, E(\|/) is not a A-adic form because it has a singularity at X = 0. However XE(\j/) is a A-adic form. Proof. Assume that k = a mod cp(p) for the Euler function (p. Write

Ek(z) - ^

X~

Then we have l a0 = (lp^^CClk)^ (l-p^^CCl-k)^ = (\^(\-^-l)L(\kM?(i>*)l2 )L(\-k,M?(i>*)l2 = Av (0;u k -l).

7.1. p-adic families of Eisenstein series

199

As already seen, an = crk_i(n) = A v (n;u k -1) if n is prime to p. When n is divisible by p, then = A v (n;u k -1). This shows the identity of the power series as in the lemma. p k " 1 E k (pz) = E k | k [ 0

Note that

A. Thus Ek(z) - p^EkCpz) is a modular form for OVi

(p

One sees easily that T contains Fo(p). The general case of non-trivial character %co"k is much easier. In fact, writing s(d) = s((d)), we see from %(u)s^d) = %((u» s(d) = X«d» that

Av(n;x(u)uk-1) = Xd-V(d)(l+x(u)u k -l) s(d) = X x ^ d ^ " 1 = ok.1)Xt0-k(n). 0
0
As seen in Theorem 3.5.2, if %co" is non-trivial and % I \i ~ V» w e (2b) A v (0;x(u)u k -l) = 2"1L(l-k,xco"k).

nave

This shows the assertion in the general case. Strictly speaking, we have only proven the proposition when k > 2 because the Fourier expansion of the Eisenstein series was not yet computed for k = l and 2 in §5.1. This will be done in Chapter 9. Now we can produce many p-adic families of cusp forms by using E(\j/). In fact, we take a modular form f e Mm(ro(pap),%;0) for a character % having values in O and make the product fE(\j/)(X) inside A[[q]]. Then we have fE(\|/)(uk-l) e f^kH Lemma 1. For u e Ox and v e O with I v | p < 1, the A(X) \-> A(uX+v) gives a ring automorphism of A = O[[X]].

substitution

Proof. For A(X) = £ " = Q anXn, we see that

A(uX+v)= S i c X-{£:=m a ^ v - Q } . The inner infinite sum is absolutely convergent p-adically, since v is divisible by p. Thus A(uX+v) is a well defined power series. The substitution of u^X-u^v for X gives an inverse map, and hence A(X) h-» A(uX+v) is a ring automorphism. Let f e ^ m ( F 0 ( p p a ) , x ; O ) . Expanding ffi(\|/)(X) as £1=0 a n (X)q n and defining a new series (called the convolution product of f and E(\|/)) by

200

7: A-adic forms, Rankin products and Galois representations

F(X) = f*E(\|0(X) = X r = o an(%-1(u)u we know that F(u k -l) = fE(\|/)(x" 1 (u)u k - m -l)e ^k(ro(pap),XoVCO-k) for all k > m , where %0 is defined by writing % as the product e%0 for characters e of W and %Q °f M- I*1 particular, if f is a cusp form of level p, we obtain A-adic cusp forms in this way. So far, we have constructed A-adic Eisenstein series using Ek(\|/). We now want to do the same thing using Gk(\|/). This is easier in fact, because Gk(x) does not have a constant term usually (see Proposition 5.1.2). We then define (3a)

B

\|/( n ; x ) = S 0(dp)=1 \|/(n/d)A d (X) for n prime to p

and G(\|/)(X) = £1=1 B¥(n;X)qn. Then, when % | p. = \|/, we have Bv(n;%(u)uk-1) = G^a-kin), and we have another p-adic analytic family of modular forms G(\|/) such that, for each character % modulo p a p, (3b) = Gk(z;x©-k)-pk-1Gk(pz;x©-k) e fA4(ro(pa+1p),coa-k) if k > 1.

§7.2. The projection to the ordinary part In this section, we first define the idempotent attached to T(p) which gives the projection to the "ordinary" part. We have defined for each character X : ( Z / p a Z ) x -> Ox

5k(r0(pa),x;o) =

%

We may regard these spaces as the O-linear span of f^k(ro(pa),X»Z[x]) or A(ro(pa),X;Z[x]) in 0[[q]]. Then the Hecke operators T(n) act on these spaces and satisfy the formula (5.3.5) describing their effect on coefficients of qn. In particular, we have the O-duality between the Hecke algebras and the spaces of cusp forms. When x is primitive modulo p a or F = SL2(Z), the Hecke algebra Hk(ro(pa),x;Q(X)) i s semi-simple and hence

H k (r,x;Q P (x)) = Hk(r,x;Q(x))®Q(%)Qp(x) is again semi-simple. Thus Hk(F,x;Qp(x)) = IIxQp(^), where X runs over conjugacy classes in HomQp(%)_aig(Hk(r,x;Qp(x)),Qp) under Gal(Q p /Q p (x)).

7.2. The projection to the ordinary part

201

Proposition 1. Let K be a finite extension of Qp(%). If f is a common eigenform of all Hecke operators in #4(ro(pa),%;K) normalized so that a(l,f) = 1, a then f is actually a complex common eigenform in Proof. We define a K-algebra homomorphism X : Hk(ro(pa),%;K) -» K by f | h = k(h)f. Then a(n,f) = k(T(n)). Since we have Hk(r0(pa),%:K) = Hk(r0(pa),%:Q(%))(E)K, we can restrict X to Hk(r0(pa),%:Q0c)). Since Hk(r0(pa),%:Q(%)) is of finite dimension over Q, X in fact has values in Q. Since H k (r 0 (p a ),x:C) = Hk(r0(pa),%:Q(%))®C, we can extend X to a C-algebra homomorphism of Hk(ro(pa),%:C) into C. Then by the duality, we can find f in ^ k ( r 0 ( p a ) , % ; C ) such that a(n,f) = ?L(T(n)) = a(n,f). Thus f = f e fAfk(r0(pa),%;C). Lemma 1. Let K be a finite extension of Q p and O be its p-adic integer ring. For any commutative Oalgebra A of finite rank over O and for any x e A , the limit lim x11" exists in A and gives an idempotent of A. n—»°

Proof. First assume that A is the p-adic integer ring of a finite extension of K. Let p be the maximal ideal of A and write pf for #(A/p). Then we have #((A//) x )=p r " 1 (p f -l). Therefore for any x e A x , x P 6 ^ s 1 m o d / . This shows that the limit of {xpfh^pf"^} as n ^ o o exists in A and is equal to 1 for x e A x . Therefore lim x n! = lim x ^ ^ ^ = 1 for x e A x . When x is in n—><»

n—><x>

p, then obviously, the above limit vanishes. Now we proceed to the general case. If the scalar extension A®0K is semi-simple, then it is a product of finite extensions of K and the image of x in each simple factor is contained in the p-adic integer ring of the factor. Then applying the above argument, we know the existence of the limit, which is an idempotent. If the nilpotent radical of A is non-trivial, write x = s + n in A<8>0K with s semi-simple and n nilpotent. (By a theorem of Wedderburn, this is always possible.) For sufficiently large f, lim s1^^"1) n—»°°

exists in the subalgebra 0[s] of A generated over O by s, since O[s] has no non-trivial nilpotent radical and is of finite rank as Omodule because it is the surjective image of A. On the other hand, if rf = 0, then

202

7: A-adic forms, Rankin products and Galois representation

Note that ft ")= pfr!/(pfr-i)!i!. Since p fn ! is divisible by (pfr-i)!pfr and 0
I ft ]lp< Cp~fr for a constant C independent of r. This shows

that the term 2Ji=l K p p n vanishes after taking the limit, and we know the existence of the limit e = lim x n ! = lim s11', which is an idempotent. n—>oo

n—><»

Let K be a finite extension of Qp(x)« We now define the ordinary projector e of the Hecke algebra H k(r0(pa),x;O) by e = l i m T(p) n! . We see easily that if f n—»oo

is an eigenform of T(p) with eigenvalue a, then (1) 0

1

if Ialp < 1.

We say a p-adic modular form f is ordinary if f | e = f. There are examples of ordinary forms and non-ordinary forms. For any character % modulo p a ( a > 0 ) , we have c m , x (p) = l+x(P)P m = 1 and a'm,%(p) = Pm+X(p) = PmThus Gk(%) I e = 0 and E k (x) | e = Ek(%) if k > 1. Now we define the ordinary part of the Hecke algebras and the spaces of modular forms by d

= eHk(r0(pa),%;O), h° k rd (r 0 (p a ),x;O) = eh k (r 0 (p a ),x;O),

= ^k(r 0 (p a ),x;o) I e, 5^rd(r0(pa),x;o) = 5 k (r 0 (p a ),x;o) I e. By definition, H k r d (ro(p a ),x;^) ^s t n e largest algebra direct summand of Hk(To(ipa),%;0) on which the image of T(p) is a unit. Again by definition, = a(l,f|hh') =
One of the fundamental theorems in the theory of ordinary forms is Theorem 1. Suppose k > 2. Then we have

rank ol^rd(r0(pa),X0)-k;O) = = ranko52ord(r0(pa),XCB"2;O).

7.2. The projection to the ordinary part

203

Here % is any character modulo p a primitive or imprimitive. The proof of this fact will be divided into two steps: (i) the first step is to show that the rank as above is bounded independently of k, and (ii) the second step is to show that the rank is equal for all k. Putting off the second step to the next section, we first prove the boundedness of the rank by cohomological means. Since Hk(r0(N),%;A) is a residue ring of Hk(Ti(N);A) by (6.3.2b), the boundedness of the rank follows from that of H k ( r i ( N ) ; Z p ) . Since H k ( F i ( N ) ; C ) = 0 x H k (r o (N),x;C) is the C-dual of M&iQi)) = e x ^ k ( r o ( N ) , x ) , by (6.3.2b), it is sufficient to prove the assertion for sufficiently large N. Thus the boundedness follows from Theorem 2. Let N be a positive integer prime to p. Then the integer rank Zp (h° rd (ri(Np a );Z p )) is bounded independently of k * / k > 2 and a > l . Proof. Write T for r i ( N p a ) . Let L be the intersection of the image L' of H ^ L f a Z ) ) in H ^ L f o R ) ) with Hj>(I\L(n;R)). Then L is a lattice of Hp(F,L(n;R)), and h k (F;Z) for k = n+2 is by definition a subalgebra of Endz(L) which is free of finite rank over Z. Let Lp = L ® z Z p . Then h k (r;Z p ) = hk(r,Z)zZp is a subalgebra of End Zp (L p ). Thus h£ rd (r;Z p ) is a subalgebra of Endzp(eLp) for the idempotent e attached to T(p). Thus what we have to prove is the boundedness of the rank of eLp independent of n. The exact sequence of F-modules 0 -> L(n;Z) ^> L(n;Z) -> L(n;Z/pZ) -» 0 yields the cohomology exact sequence

Therefore H 1 (r,L(n;Z))®Z/pZ can be embedded into H^rjLfaZ/pZ)). Note that L/pL = Lp/pLp, L/pL injects into L'/pL1, and L'/pL' is a surjective image of H1(T>L(n;Z))/pH1(r,L(n;Z)) = H1(r,L(n;Z))(8)zZ/pZ. Thus it is sufficient to show that the dimension of eH ! (r,L(n;Z/pZ)) is bounded independently of n. We shall show this fact by constructing an embedding of eH^rjLfaZ/pZ)) into e H ^ Z / p Z ) . Note that, for P(X,Y) = I ^ a i X ^ Y 1 e L(n;A),

x

J

P(X,Y) = £ a^X-mYf-V andhence i=o ^

P (1,0) = P(l,0). ^ ^

204

7: A-adic forms, Rankin products and Galois representation

Let us define maps i : L(n;Z/pZ) —> Z/pZ and j : Z/pZ —» L(n;Z/pZ) by i(P(X,Y))=P(l,0)

and

j(x) = xY n . Since Y = L

j mod p

for any

y e r = ri(Np a ), i and j are homomorphisms of F-module. Thus combining i or j with a 1-cocycle u, we obtain the following two morphisms of cohomology groups: I = i* : H^rjLfoZ/pZ)) -» H ^ Z / p Z ) and j * : H ^ F , Z/pZ) -> H ^ F , L(n;Z/pZ)). We want to show that I is an isomorphism of oH1 (T,L(n;Z/-pZ)) ^ ) . We consider the exact sequence of F-modules

onto

0 -> Ker(i) -» L(n;Z/pZ) -» Z/pZ -> 0. This yields another exact sequence: H^r.KerCi)) -» H ^ r . L f a A ) ) -> H ^ Z / p Z ) -> H 2 (r,Ker(i)). (1 (A Note that for a = L.

t

La

leaves Ker(i) stable and hence T(p) acts natu-

rally on Hq(F,Ker(i)). On the other hand, the Z/pZ-module Ker(i) is generated by monomials X ^ Y 1 for i > 0 . Thus for a = nilpotenton Ker(i). Since r a r = U£ d 1 ra

, the action of a 1 is

, the action of T(p) is nilpo-

tent on Hq(F,Ker(i)) for q > 0. This shows the desired assertion. We shall give another proof of the theorem when a = 1. When a = 1, we modify j * and construct a map J : H^F, Z/pZ) -» H ! (r, L(n;Z/pZ)) so that J°I = (-l)nT(p) on H^FiCNp), L(n;Z/pZ)). Thus (-lfTip)'1] actually gives the inverse of I on eH^F^Np), Z/pZ). We consider the double coset F8F for 5 E SL 2 (Z) such that (0 - n 8 = L 0 j mod p and 8 = 1 mod N. One can always find such an element (see Lemma 6.1.1). Then we have a disjoint decomposition

r s r = Uf 1 r55i ** % = (J J). Similarly we take x e M2(Z) with det(x) = p such that

0 J mod P

and T

^ [ o p j m o d N'

7.2. The projection to the ordinary part

205

We can find such x as follows. By Lemma 6.1.1, we can find a e SL2(Z) such that G = 1 mod p 2 and 0 -\\ mod N 2 . a s Then x =

Q

a does the job. Then one can easily verify that x normalizes

F and induces an automorphism of H^I^Z/pZ) = Hom(F,Z/pZ) which takes u : F -^ Z/pZ to u | [x](y) = uCxyx"1). Note that

because FxSSi =

0

mod Np

and det(x88i) = p. Define

J to be

[r8n o j*°fr3. We compute J°I. Let u : F -> L(n;Z/pZ) be a 1-cocycle. Then if 5iY = Yi55j for y,yi e F, then xSSiy = xyiX^xSSj with xyix"1 e F. Thus

Note that j(i(P(X,Y))) = a0Yn = (-l)nxlP(X,Y). Thus yiX"1) = (-l)nu I T(p). Thus, we have J°I = (-l)nT(p), giving another proof of the theorem. Here we add one more result for our later use in §10.4, which is a special case of [HI, Th.3.2]: Proposition 2. If k > 2 and p > 5, then e induces e : ^ rd (SL 2 (Z);Q p ) = 4 rd (r o (p);Q p ), e : f^rd (SL2(Z);Qp) = ^ r d (T0(p);Qp), where the ordinary part for SL2(Z) is defined with respect to the Hecke operator T(p) of level 1. Proof. We start a general argument. We consider any T = F(N) = {y e SL 2 (Z) | y = 1 mod N} for N > 3. Then F is torsion-free. Now we put Fo = FflFoCp). Then we have I1 = lores: H ^ F ^ n j Z / p Z ) ) -+ ^(ToM^Z/pZ))

-> H

Since we can show by the strong approximation theorem (Lemma 6.1.1) that r

(o p) r = U o s u < p r (o T ) U r a l o i)

(see [Sh

'

206

7: A-adic forms, Rankin products and Galois representation

for a G SL 2 (Z) with a =

mod N, we know that

V ° V)

°Yl l

,

1 J|P)

l

fP

0>

l

= (a P)((l,0)[0 J) = (alP)(0,0) = 0 (if n>0)

for any homogeneous polynomial P with coefficients in Z/pZ. Since T(p) of level N and T(p) of level Np are different, we add the subscript "N" to indii i i fP 0>l cate the level. Then f | TN(P) = f I TN P (P) + f I ol Q X I. By the above argument, the term corresponding to a I Q A does not affect to the value of i*. Similarly we consider the trace map as in §6.3:

Tr = [r 0 l 2 r] : HHroJL^Z/pZ)) -> HHlM^ Then we consider j ' = Troj^ox : H^roJLCO.o^Z/pZ)) -> H^roJLfoZ/pZ)) -» Then basically by the same computation as in the case of a = 1, we have J'or = (-l) n T N (p). Let us now compute I'OJ'. We pick a cocycle v e Z(r0,L(0,co"n;Z/pZ)). Then, noting that T = IIo
u=0

where 88 u y = y u a for an a in {88V I v = 0,...,p-l }U{ I2} for the identity matrix I2. Here we have used the fact that i°j = 0. By this, we have (2)

H^ rd (r,L(n;Z/pZ)) s H^rd(r0,L(0,co-n;Z/pZ)) = H^ rd (r o ,L(n;Z/pZ)).

Now we consider the cohomology sequence attached to 0 -> L(n;Z p ) —^-» L(n;Z p ) -> L(n;Z/pZ) -> 0, which gives rise to, for O = T and To, (3) 0 -> Hi(O,L(n;Zp))®zZ/pZ -> H^O.UnjZ/pZ)) -> Hi+1(O,L(n;Zp))[p] -> 0, where H 2 (O,L(n;Z))[p] is the kernel of the multiplication by p on H2(
H^OJL

^

7.2. The projection to the ordinary part

207

Note that H°(O,L(n;Z/pZ)) = L(n;Z/pZ)^ may be non-trivial. We have the map induced by the projector e attached to T(p) of level p: e: Then (2) shows that e is an isomorphism after reducing modulo p. Then by Nakayama's lemma, we know that the map e is surjective. If x is an element in I (\ NuY x. We know that for each monomial L(n;Z/pZ), then x | T(p) = Xo
which is obviously 0 if j > 0 because p = 0 in Z/pZ. Thus, we know (I NuY n ^

cons

* s t s on^y of terms involving Y.

This shows x | T(p) 2 = 0 and hence H° rd (O,L(n;Z/pZ)) = 0 because e = limT(p)n!. Applying the operator e to the exact sequence (3) for i = 0, we have H^ rd (O,L(n;Z p ))[p] = 0. Thus Ho rd (0,L(n;Z p )) is torsion-free. Therefore we conclude from (2) and (3) that e induces an isomorphism: (5)

H^ rd (r,L(n;Z p )) = Hi rd (r 0 ,L(n;Z p )).

Note that G = SL 2 (Z)/r = r o ( p ) / r o = SL 2 (Z/NZ). If N is a prime, then #(SL 2 (Z/NZ)) = N(N+1)(N-1). Since p > 5, we can always choose N so that #(SL2(Z/NZ)) is prime to p. Then it is well known that res: H 1 (SL 2 (Z),L(n;Z p )) = H°(G > H 1 (r,L(n;Z p ))), res: H 1 (r 0 (p),L(n;Z p )) = H 0 (G f H 1 (r 0 ,L(n;Z p ))). This combined with (5) shows that e induces H1Md(SL2(Z),L(n;Zp)) = H^ rd (r o (p),L(n;Z p )). By this, we know that d i m ^ r d ( r o ( p ) ; Q p ) = dim^ r d (SL 2 (Z);Q p ), which shows the assertion for f&4- Then the assertion for 5k follows from that for M^.

208

7: A-adic forms, Rankin products and Galois representation

Corollary 1. For all fields A of characteristic 0, the algebra h£rd(Fo(p);A) is semi-simple if k > 2 and p > 5. In fact, the assertion is true even for k = 2. This fact follows from the fact that 5 2 (SL 2 (Z)) = 0 and [M, Th.4.6.13]. In this case, if f e 5 2 (r 0 (p)) is a normalized eigenform, then f is primitive in the sense of [M, §4.6] and f

|

x = ± p

(

k

-2)/

2 f

c

f

o

r

x

=

^

" ^

Proof. We know that hk(SL2(Z);C) is semi-simple. Thus h°rd(SL2(Z);Qp) is semi-simple. Thus we can find a basis {fi, ..., fr} of 5£ rd (SL 2 (Z);Q p ) consisting of common eigenforms of all Hecke operators. Let f be one of them. We write f | T(p) = af. Then | a I p = 1. We take roots a and p of X 2 -aX+p k " 1 = 0 . Then one of a and p, say a, is a p-adic unit, i.e. I oc | p = 1. Then | p | p = p 1 " k < l . We define f = f(z)-pf(pz). Then it is easy to check by the formula (5.3.5) that f I T(n) = a(n,f)f if n is prime to p and f | T(p) = af. Thus f is an ordinary form. As shown in the proof of Theorem 5.3.2, if a(q,fi) = a(q,fj) for all primes outside p, then i = j . This implies fi', ..., fr' are linearly independent. Then by Proposition 2, they form a basis of 5k r d (r o (p),Q p ). Therefore h£ rd (SL 2 (Z);Q p ) is semi-simple, which shows the assertion.

§7.3. Ordinary A-adic forms In this section, we study the structure of the space of ordinary A-adic forms following the method of Wiles [Wil]. Actually the space of A-adic forms is the A-dual of the p-ordinary Hecke algebra of level p°° defined in [H3] and [H4]. Then all the results concerning the structure of the space of ordinary A-adic forms follow from the structure theorem of the ordinary Hecke algebra proved in [H3] and [H4]. However, we have adopted the method of Wiles, which is more compact. We ease (in appearance) a little bit the requirement (A) to be a A-adic form given in §7.1: for each character % modulo p a p (which may not be primitive), a formal q-expansion F(X;q) = Z°° a(n;F)(X)qn with coefficients in A = O[[X]] n=0

is called a A-adic modular form F(X) of character % (with values in (?) if the following condition is satisfied: for the generator u = 1+p (A1)

F(u k -l;q)e #4(ro(p a p),xco' k ;0) for almost all positive k (i.e. all but finitely many positive k).

7.3. Ordinary A-adic forms

209

When F(u k -l;q) is a cusp (resp. a p-ordinary) form for all sufficiently large k, we say that F is a cusp (resp. an ordinary) form. Let M = M(%;A) (resp. S = S(%;A), M o r d = M ord (%;A), S o r d = S ord (x;A)) be the A-module of all A-adic modular (resp. cusp, ordinary modular, ordinary cusp) forms. To introduce a Hecke operator on M and S, we consider the character x S K : W = 1+pZp -> A given by K(US) = K(U S )(X) = (1+X) . It is obviously a continuous character with respect to the m-adic topology on A, where m is the maximal ideal of A. Note that for integers n prime to p, K((n))(uk-1) = K(us(n))(uk-l) = u ks(n) = 0)-k(n)nk, where we write (n) = co(n)-!n = u s(n) (s(n) = log((n))/log(u)). Then we define for each A-adic form F e M(%;A) a formal q-expansion F | T(n) by (1)

a(m,F | T(n))(X) = £ b | ( m n ) K«b))(X)%(b)b-1a(mn/b2,F)(X),

where b runs over all common divisors prime to p of m and n. We evaluate this formal power series F|T(n) at u k -l where F(uk-l;q) is meaningful as a modular form. Then we see that a(m,F | T(n))(uk-1) = £ b | ( m n ) K((b»(uk-l)%(b)b-1a(mn/b2,F)(uk-l) = Xb|(m,n) Xa)- k (b)b k "Vmn/b 2 ,F(u k -l)) = a(m,F(u k -l) |T(n)). This shows that F | T(n)(u k -1) = F(u k -1) | T(n) e ^ k (r 0 (p a p),%co- k ;0). Therefore, F is again a A-adic form. Thus, the operator T(n) is well defined and so we now have Hecke operators T(n) acting on M and S and their ordinary parts. Lemma 1 (Weierstrass preparation theorem). Any power series F(X) in A can be decomposed into a product of a unit power series U(X), some power of a prime element in O, and a distinguished polynomial P(X) e O[X]. (A polynomial P(X) = ao+aiX+---+X n is called "distinguished" if | a i l p < l for all i.) Since this fact can be found in any book in commutative ring theory or p-adic number theory (for example [Bourl, III], [L, V.2], [Wa, Th.7.3]), we omit the proof. By this lemma, each non-zero power series with coefficients in O has only finitely many zeros in the disk { x e O

| x | p < 1}.

Theorem 1 (A. Wiles). The space of ordinary A-adic modular forms (resp. ordinary A-adic cusp forms) of character % is free of finite rank over A.

210

7: A-adic forms, Rankin products and Galois representations

Proof. The proof is the same for M ord and Sord. We shall give a proof only for M ord . We prove first that M ord is finitely generated and is A-torsion-free. By definition, M ord is a A-submodule of the power series ring A[[q]]. Therefore it is A-torsion-free. We now prove that the rank of any finitely generated free A-submodule M of M o r d is bounded. Let {F b F 2 , ••• , Fr} be a basis of M over A. Since Fi, "-.Fr are linearly independent over A, we can find positive integers ni, ..., nr such that D(X) = det(a(nj,Fj)) * 0 in A. By the above lemma, we can take the weight k so that D(u k -l)*0 and Fi(uk-1) has meaning, that is, is an element of f^£rd(ro(paP)>%G)'k;0) for all i. Write £ for Fi(uk-1). Then D(uk-1) = det(a(ni,fj)) * 0. Thus the modular forms fi, ..., fr span a free module of rank r in fAf£rd(ro(pap),%co'k;0) whose rank is bounded independently of the weight k. Thus r is bounded by a positive number independent of M. This shows that if Fi, • • •, F r is a maximal set of linearly independent elements in M ord , any element in M ord can be expressed as a linear combination of the Fi's if one allows coefficients in the quotient field L of A. We thus consider V = M ord ® A L, which is a finite dimensional space over L embedded in L[[q]]. For each F e M o r d , write F = Si xiFi with xi e L. Then xi is the solution of the linear equations (a(ni,Fj))x = (a(ni,F)) e A r . Therefore Dxi G A, and thus DM ord is contained in AFi + ••• + AFr. Therefore M ord is finitely generated since A is noetherian. To prove the freeness over A, we note the following facts: (i) (ii)

A is a unique factorization domain; A is a compact ring.

The first fact follows easily from Lemma 1 (see [Bourl, VII.3.9]). The second fact follows from the fact that for the maximal ideal m of A, A/ni1 is always a finite ring and we have topologically A = lim (A/ma) = lim (A/P a ) for any non-trivial element P in nu Since M ord is finitely generated, we can find k so that F(u k -1) is meaningful for all F in M o r d . If F(uk-1) = 0, then a(n,F)(u k -l) is divisible by P = P k = X-(u k -l) for all n. Thus by dividing F by P, we still have an element of M ord , because (F/Pk) = F(uj-l)/(uj-uk) for all j & k, which is a modular form. Thus P M o r d = { F G M o r d | F(u k -l) = 0}. So M o r d / P M o r d can be embedded into ^ r d (r 0 (p a p),%0)- k ;O). Thus M o r d / P M o r d is O-free of finite rank. Let us take Fi (i=l, — , r ) so that FimodPM o r d gives an Obasis of M ord /PM ord . Note that the Fi's are linearly independent over A. In fact, if not, we may suppose that A4F1+ ••• +A,rFr = 0

7.3. Ordinary A-adic forms

211

with at least one of the Xfs not divisible by P. Then reducing modulo P, we have a non-trivial linear relation between the Fi mod P, which is a contradiction, and hence the Fi's are linearly independent. Consider M = AFi+ ••• +AFr. Then M is a A-free module of rank r and M/PM coincides with M ord /PM ord because if F is an element of M ord , then we can find a finite linear combination Go of the Fj's such that F-Go is divisible by P. We now apply this argument to (F-Go)/P and get another linear combination Gi of the Fj's such that (F-Go)/P-Gi is divisible by P. Continuing this process, we can find the Gj's which are linear combinations of the Fj's such that F = G0+G1P+ ••• H-Gj.iPJ-1 mod P j . Thus M/PjM = M ord /P j M ord . Note that the series G0+G1P+ ••• H-Gj-iPJ"1 converges in M by identifying M with Ar by the basis Fj's. Thus M = M ord and hence M ord is A-free. From the above proof, we know for sufficiently large k that M ord /PM ord for P = X-(u k -l) is naturally embedded into ^ rd (r 0 (p a p),%co" k ;0); in particular, we have, for sufficiently large k, rankA(Mord(%,A)) < rank 0 (^ r d (ro(p a p),xco" k ;0)). Now we want to define the idempotent e on M = M(%,A). Take F in M. We may assume that F(uk-1) is meaningful for every k > a . We consider the sum a*a,k(A) = I 3 k =a ^j(ro(pap),%co"j;A) inside A[[q]] for a subalgebra A of C or Q p . The space fAfa,k(A) is in fact isomorphic to the direct sum. In fact, over A which is a subalgebra of C , we see that if Z j = a fj = 0 for fjG fAfj(r0(pap),xco"j;A), then for any ye Tx(pap)9

'j=a

= S j = a fj(z>J(Y'Z)J = °-

We can of course choose j[ e r i ( p a p ) so that det(j(Yi»zy) * 0 (0 < i < k-a and a < j < k ) . This shows that fj = O. For A inside Q p , the space over A is defined by the scalar extension of the space over QflA; hence, we know the assertion. Thus we have a natural action of T(n) on fAfa,k(A). Since M^iO) is O-free of finite rank, the subalgebra R of Endo(^ a ,k(^)) generated by T(p) over O is Ofree of finite rank. Thus we can take the limit ek = lim T(p) n! in n—>«*

R. This idempotent preserves fWa,j(0) for j < k and induces the projector e on each subspace stable under e. Now we consider the A-submodules M a , k = { G e M I G(u j -l)e 5Wj(ro(pap).XCO-J;0) for all j in [a,k]}, M ' a , k = { G e M a , k I G(u j -l) = 0 for all j in [a,k]}.

212

7: A-adic forms, Rankin products and Galois representations

We note that M'a>k may not equal £*kMa,k> where Q^ = Il]La(X-(u:'-l)). By definition, the map G f-» £ k =a G(u J -l) induces an injection of Ma>k/M'a,k into Since T(p) preserves M'a>k and Ma,k, the idempotent ek acts on Ma)k/M'a)k. If a < j k/M'a>k to M a yM' a j satisfying ej°7ik,j = ^k,j°ek- Note that Q^A is the kernel of the map G i-> Z- a G(uM) on A, and hence Ma?k/M'a>k can be embedded into (A/Qk)[[q]L The sequence of ideals {^kA} is a decreasing sequence of ideals of A and f l ^ k A = {0}, since any power series has only finitely many zeros inside the unit disk in Q p . Thus A = lim (A/Q k ). Therefore a(n,F|e)(X) = lim a(n,F | ek) exists in A and a(n,F | e)(u k -l) = a(n,F(uk-l) | e). This shows that F | e e M ord . Since the projective limit topology of the p-adic topology of A/Qk coincides with the m-adic topology of A for the maximal ideal m of A, the above proof shows (2)

F | e = KmJF | T(p)n!) under the /rc-adic topology.

Summing up, we have Proposition 1. There exists a unique projector e on M(%,A) to Mord(%,A) which satisfies F | e(u k -l) = F(uk-1) | e for all F e M(x,A). In §1, we constructed a A-adic modular form E(%) out of Eisenstein series for characters % of [i ([i = {£ e Z p | ^P- 1 = 1}). We can extend the construction to characters % modulo p a p as follows. Let % : (Z/p a pZ) x -^ cf be a primitive character, and put %o = % I p.- We define E(%)(X) = E(%o)(£X+(C-l)) for £ = x(u). Then we see from Proposition 1.1 that E(%)(uk-1) = E(%o)(£uk-1) = Ek(%G)"k). Thus E(%) is a A-adic form of character % in the sense of (A'). Moreover take a modular form g e fMa(ro(p^p),\|/;0) for a finite order character \\f : Z p x -^ Ox, and put \\f0 = \\f \ ^. Then multiplying ECxyo-1) to g and making the variable change X h-» \|/(u)"1u"aX+(\)/(u)'1u"a-l) in gE(%)(X), we obtain g*E(%\j/0-1)(X) G M ( % \ J / 0 , A ) such that g*E(%\|/ 0 " 1 )(u k -1) = gEk-a(%V"lco k) for all k > a, which is an element in #4(ro(pYp)>%a>"k;0) for y = max(a,(3). For any A-algebra A, we define Mord(%;A) = Mord(%;A)®AA. Now we prove Proposition 2. Let % be either a primitive character modulo p a p having values in Ox or % = id and a = 0. Let g = G a (\|/) for a character \\f : ( Z / p a p Z ) x -> Ox with \j/(-l) = (-l) a . Then the elements of the form

7.3. Ordinary A-adic forms

213

(g*E(x\|f0-1) I e) | T(n) (Vo = v l ^ together with E(%) span Mord(%,L) over the quotient field L of A. Moreover if k > 2a+2 tfnd %cok w primitive modulo p a p, ?/*£« ?/*e K-subspace of K[[q]] spanned by the specialized image of M ord (x,A) at weight k contains the whole space M™d (ro(pap),%co~k;K) and coincides with f^£rd(ro(pap),%co'k;K) if k is sufficiently large. Taking a to be 2, we know by Corollary 5.4.1 that any ordinary modular form can be lifted to a A-adic ordinary form (up to constant multiple) if k > 6 and that Mord (x,A)/P k M ord (%,A)(8)K= ^ r d (r o (p a p),%co- k ;K) if k is large and %Grk is primitive. Moreover we know the independence of the dimension of ^ k r d (ro(PaP)>%co~k;K) from the weight if k is large, as long as %ccrk is primitive. This fact is actually true for all k > 2 as we will see later. Proof. By Theorem 5.4.1, as long as %cok is primitive, the modular forms of the type g * E ( x w 1 ) I T(n)(u k -1) = g E ^ t y ^ C D ^ ) I T ( n ) ( n = l , 2 , - ) together with Ek(%co-k) = E(%)(uk-1) and Gk(%Grk) span f*4(F,%co'k;K) for F = Fo(p a p). Hence ordinary forms of the form (g*E(%\1/0-1) I e) I T(n)(uk-1) = (gEk.a(%\|/-ico-k) | e) | T(n) together with E(%)(uk-1) span M™d (ro(pap),%co"k;K), because we know that e annihilates Gk(%ork) because Gk(%ork) | T(p) = pk~1Gk(%Grk). Thus the subspace of K[[q]] spanned by the specialization of A-adic forms at weight k contains f^krd(ro(pap),XCO'k;K) if k>2a+2. By the proof of Theorem 1, it is clear that for sufficiently large k, the specialization map from M/P k M (Pk = X + l - u k e A) for M = Mord(%;A) into ^ rd ( r o(P a P) 5 XW" k ;K) is injective. Choosing large k so that %ark is primitive, we then know that {Fi} ie i (in the proof of Theorem 1) span M o r d ( % ; L ) if { F i ( u k - l ) h span rd a k fAf£ (ro(p p),%co~ ;K) over K. This shows the proposition. By definition, we also know that, for any character e : W/Wp^ —»

(g*E(%W11 e) I T(n)(e(u)uk-1) = (g*E(xw 1 )(e(u)u k -l) I e) I T(n) = (gEk.a(e%T1co"k)le)|T(n).

Defining formally e^F(X;q) = F(e(u)X+(e(u)-l)) as an element of A[[q]], we see from the above formula that e*(g*E(%\|/o"1) I e) I T(n) is a A-adic form of character £% as long as e% is primitive modulo ppmax(a'P). We now show that this is always the case without assuming the condition on primitivity. The point of the argument is that for integer P > a, defining Fa?p = Fi(p a )nro(p^), we have

214

7: A-adic forms, Rankin products and Galois representations (I

0)

J

r

r

This fact can be shown as follows.

|

(I (A

Jr

sets

-

Writing r\ for

(I

ff\

0

, we know

m r = UiFriSi is disjoint if F = UiOl^rrinnSi is a disjoint union. In our case, we see easily that

U t ^ for the same

8i =

and ft

U

L This implies that

^

T(p)^"a

^ not only acts on

A:(ro(p^p),%) but also decreases the level up to p a p if % *s primitive modulo p a p. In other words, T(p) p - a brings 5k(r0(pPp),%) into 5 k (r 0 (p a p),%). Thus actually £ * ( g * E ( x w ! ) I T(n) | e) = E*(g*E(%y0-1) I T(n)) | e has level p a p, which is the conductor of e% (i.e. e% is primitive modulo p a p). This shows our claim. Thus F H> e*F takes ordinary A-adic forms of character % to those of character £%, because the forms g*E(%\|/o"1) I T(n) | e (n = 1, 2,...) span the total space Mord(%;L). Since e* is induced from the ring automorphism of the ring A, e* is injective. Since (££')* = £*£'* and hence

E^E'1*

= E'1^^

= id* = id,

ord

£* has to be a surjective isomorphism. We have £* : M (%;A) = Mord(£%;A) for any finite order character £ : W —> (?. Summing up we have Theorem 2. For each finite order character £ : W —> O*, we have an isomorphism £* : M ord (%;A)= Mord(£%;A) functorial in E (i.e. (££')*=£*£'*;. Moreover, for almost all positive integers k, F(£(u)uk-1) is an element of ^krd(Fo(pPp),e%co"k;0) if F e Mord(%;A), where p is the minimum exponent such that E% factors through WAVP . The same assertion also holds for cusp forms. By this theorem, without losing much generality, we may assume that % is a character modulo p as we did in the definition (A). Hereafter we assume that % is a character of (Z/pZ)x . For each character £ of W with values in Q p , we write O[£] for the subring of Q p generated over O by the values of £. To prove the following theorem, we need a careful analysis of group cohomology attached to the space of modular forms. Although the following theorem itself is true for all primes p, the cuspidal theory is empty for p < 7 because a posteriori we find that Sord(%;A) = 0 for all p < 7. (We should mention that if we consider Fo(Npa) in place of Fo(pa) for N prime to p, the theory is not empty even for p = 2 and 3.) Assuming p > 5, the analysis of group cohomology becomes a

7.3. Ordinary A-adic forms

215

lot easier because Fi(p) is torsion-free. For this reason, we throw away the primes p = 2 and 3 and give the exposition only when p > 5. T h e o r e m 3 . For all integers k > l and a primitive finite order character e of W/WpCl and for any modular form f e f5Vfk(r0(pap),e%co"k;0[e]), there exists a A-adic form F of character % such that F(e(u)uk-1) = f. Moreover if k > 2, we have the isomorphisms: d k Mord(x;A)/Pk,eMord(X;A) = f Sord(x;A)/Pk,eSord(x;A) = 5 where the map is induced by Fb->F(e(u)uk-l) and Pk,e is the prime ideal generated by X-(e(u)uk-l). Proof. We consider the A-adic Eisenstein series E' = XE(id)(X;q). We know from Theorem 3.6.2 that

rV-DCpd-s) I s=0 =

r'logiu^-i)

which is a p-adic unit. Thus we have E'(0;q) = 2~1/
g*E(X;q) = gE(el(u)ukX+(e\u)uk-l))

E M(X;A),

we have (g*E) | e(0) = (g*E(0)) I e = g. This shows the first assertion. By Proposition 2, we get, if k is sufficiently large (*) {Mord(%;A)/Pk,eMord(x;A)}(8)oK =

^

£

Since the unique Eisenstein series in f^krd(Fo(pap),exco"k;K[e]) is given by E(%)(e(u)uk-1), we conclude, if k is sufficiently large, that (**)

{SordOc;A)/Pk,ESord(x;A)}®0K

s

5™ = j j

This combined with the first assertion shows the second assertion for large k. Thus we need to show that (*) and (**) hold for k > 2. First we show that (*) for k > 2. Then (**) follows from (*) by the same argument as above. Since every classical (ordinary) form lifts to a A-adic ordinary form, the image M of specialization in O[e][[q]] contains f^r£ and M/fAfkr is O-free. Thus we only need to prove the following equality of the ranks: l

= rank A M ord (x;A).

216

7: A-adic forms, Rankin products and Galois representations

For this we may extend scalars and may therefore assume that e%co"k has values in 0. Let \j/ : (Z/p^pZ)x —» 0* be a character. We pick a prime element G3 of 0 and put F = 0/G30. We take a normal subgroup A of Fo(p) so that Fo(p)/A has order prime to p and A is torsion-free. If p > 2, A = F(4) does the job. Then for any 0fFo(p)] -module M, H°(r o (pp a ),H q (Anr o (ppP),M)) = Hq(F0(ppP),M) because Tr°res = (Fo(p):A). For the moment, we suppose that p > 2. Then by Proposition 6.1.1, H 2 (Anr 0 (pp a ),M) = 0 = H 2 (Fo(pp a ),M). From the cohomology exact sequence attached to (***)

0 -> L(n,\|/;0)—-^->L(n,\|/;0) -> L(n,\|/;F) -> 0,

we get an exact sequence for F = Fo(p^p): 0 -» H 1 (r,L(n,\|/;0))(8) o F -> H ^ r j L C n ^ F ) ) -> H 2 (F,L(n,\|/;0)) = 0. This shows that dimKeH1(r0(ppp),L(n,\|/;K)) = dim F eH 1 (r 0 (p P p),L(n,\|/;F)). Now we again consider the map i : L(n,\|/;F) —» L(O,\j/con;F) given by i(P(X,Y)) =P(l,0), which is a homomorphism of Fo(p^p)-modules. Here we have \j/con in place of \\f because on Xn,

TO(PPP) ^ 7 = fa0

°] modp for ae Z p x

acts by an\}/(a) = \}/con(a) mod p. Exactly in the same manner as in the proof of Theorem 2.2, we know that i induces an isomorphism ,\|f(Dn;F)). Note that \|/(u) = 1 mod p for the maximal ideal p of 0 because \\f(u) is a p-power root of unity and there are no p-power roots of unity except 1 in a characteristic p situation. Thus if \|/ = e%co'k, we have for T = ro(p a p), writing Xjx for the restriction of % to ja and noting the fact that % = X\i m ° d p, i # : e H ^ r ^ n ^ F ) ) = eH^F^CO^co^F)) = eH1(r0(p),L(0,Xn(0"2;F)). Here the last isomorphism follows from the fact that e decreases the level to the conductor of the character. Now, for any 0-module M, writing M[G3] for the submodule killed by C3, from the exact sequence (***), we have the following exact sequence for F = Fo(ppp):

7.3. Ordinary A-adic forms 0 -> H°(r,L(n,\|/;O))<S>oF - ^

217

H°(r,L(n,\j/;F)) -> H ^ L C n ^ O ) ) ! ® ] -> 0.

We see easily by definition, for all j , that

If j > 0, then X n - j Y j | T(p) = 0 mod p. If j = 0, X n | T(p) mod p does not have a term involving Xn. Thus X n [ T(p) 2 = 0 mod p. Anyway, we have eH°(r,L(n,\|/;F)) = 0 for all \|/ and n. Thus eH1(r,L(n,\)/;O)) is O-free. Thus we have from Theorem 6.3.3 that k

; 2dimK Mlri (ro(pap),exco--k;K)= dimKHj)rd(r0(ppc'),L(n,exco-k;K)) = dimFH^d(ro(pp«),L(n)e%co-k;F))

K) + dimK ^ 2 o r = 2dim K iW! rd (r o (p),x H co- 2 ;K)-l. This shows that dimK2tfkrd(ro(q),exco~k;K) is independent of k > 2 and e. This number coincides with the rank of Mord(%;A) because of (*) for large k. Therefore for any k > 2, we have dim K ^ ld (ro(q),exco- k ;K) = rankAMord(%;A), which is what we wanted. This also finishes the proof of Theorem 2.1 when p > 2. We now give a sketch of the argument in the case when p = 2. We replace ToCpp") in the above argument by r o ( p p a ) = r o (pp a )f]ri(5). Then r o ( p p a ) is torsion-free, and we know by the same argument that

HL(r'o(paP),L(n,exco-k;F)) s H This shows first that

and then taking the subspace invariant under ro(p a p), we can conclude with the desired identity: dimKW"rd(ro(2a+2),e%0)-k;K) = dimK< r d (ro(4), e t l o)- 2 ;K) = 1.

218

7: A-adic forms, Rankin products and Galois representations

Anyway the ordinary part Sord(%;A) = 0 if p < 7 as is clear from the fact that dim5k rd (ro(p)) = dim5k rd (SL 2 (Z)) = O if k < 1 0 (see Proposition 2.2 and the dimension formula for 5 k rd (SL 2 (Z)) in §5.2). We now define the universal ordinary Hecke algebra Hord(%;A) (resp. hord(%;A)) by the subalgebra of EndA(Mord(%,A)) (resp. EndA(Sord(x,A))) generated by all the T(n)'s over A. For any A-algebra A, we define Hord(%;A) (resp. hord(%;A)) by Hord(%;A)AA (resp. hord(x;A)®AA). Similarly we define Mord(%;A) = Mord(%;A)®AA, Sord(%;A) = Sord(x;A)
M ( x ; A ) = M(x;A)(g>AA and S ( x ; A ) = S ( x ; ^ Then we have the well defined projection map e : M(x;A) -> Mord(x;A). Theorem 4 (semi-simplicity). Hord(x;A) (resp. h ord (x;A)j is reduced; i.e., Hord(x;L) (resp. hord(x;L)j for the quotientfield I of A is semi-simple. Proof. We choose a basis {Fi}i=i,...,r of M o r d = M ord (x,A). Then we can identify End A (M ord ) with the matrix ring Mr( A).

Suppose h e Hord(x;A) is

ord

nilpotent. For k > 2 , EndA(M )(8)AA/PkA = Endo(M ord /P k M ord ) for P k = X-(u k -l). Note that the image of h in End o (M ord /P k M ord ) gives an element of Hkrd(ro(pap),XO)-k;0) because Mord/PkMord(3)oK= < r d (r 0 (p a p),xo)- k ;K). By Corollary 2.1 and Theorem 5.3.2, H^ rd (ro(p a p),xco' k ;0) has no non-trivial nilpotent elements. Thus the image of h in H£rd(ro(pap),xco'k;0) is trivial. Thus h is divisible by P k . Since we have h e f]kPkMr(A) = {0}, where the intersection is taken over all k > 2, this finishes the proof. We now define the pairing (,):Hord(x;A)xMord(x;A)->A

by
We also define m o r d (x;A) by {Fe M o r d (x;K) | a(n,F) e A if n > 0 } , where K is the quotient field of A. Theorem 5 (duality). For any extension A of A, the above pairing induces (i) HomA(Hord(x;A),A) = m ord (x;A) and Hom A (m ord (x;A), A) = Hord(x;A), (ii) Hom A (h ord (x;A),A) s Sord(x;A) and HomA(Sord(x;A), A) = h ord (x;A). In particular, Hord(x;A) and hord(x;A) are free of finite rank over A. The freeness of hord(x;A) over A was first proven in [HI] directly without using A-adic forms for p > 5.

7.3. Ordinary A-adic forms

219

Proof. We can prove the first part of (i) and (ii) in exactly the same manner as Theorem 5.3.1. Since the arguments are the same for (i) and (ii) for the second part, we only prove the second part of (ii). Since Sord(%;A) is A-free, we may assume that A = A. We note that over the integral closure A of A in a finite extension of L, for each A-torsion-free module M, the double dual M** = (M*)* may not be isomorphic to M, where M* = HOITIA(M,A). For example, for the maximal ideal m of A, //*** = A. Writing h (resp. S) for hord(%;A) (resp. Sord(%;A)), we have a natural map h -> h** = HomA(S,A) which is A-free. This map is injective because of the non-degeneracy of the pairing. Thus N = h**/h is a torsion A-module. Since after localizing at any height one prime P of A, we get the identity h** P = HomAp(HomAp(hp,AP),Ap) = h P because Ap is a discrete valuation ring. Since the Krull dimension of A is 2, N is killed by a power of m. Thus N is a finite module. Since h** is A-free, S = h***. Thus we have (*)Hom o (h**/P k h**,O) = S/PkS

On the other hand, from the exact sequence 0 -» h —> h** -» N —> 0, we get another exact sequence (see Theorem 1.1.2): Tor^(N,A/P k A) -> h/P k h -> h**/P k h** -> N/PkN -> 0. Since N is finite, the module TorA(N,A/PkA) is finite, and thus we have another exact sequence 0 -> h k (r 0 (p),%co" k ;0) -> h**/P k h** -> N/PkN -> 0, because the image of h/P k h is the subalgebra of the O-free algebra h**/P k h** generated by the T(n)'s, which is hk(ro(p),%co"k;0). The above exact sequence yields, by O-duality (see Theorem 1.1.1) 0 -> Hom o (h**/P k h**,O) -* 4rd(r0(p),%0)-k;O) -^ Ext^(N/PkN,O) -> 0. Then by (*), we know that ExtJ,(N/PkN,O) = 0. Since O is a valuation ring, ExtJ,(N/PkN,O) = N/P k N = 0

(Corollary 1.1.1). Then Nakayama's lemma

shows that N = 0 showing h = h** = Hom A (S,A). By Theorem 5, we know that Hord(%;L) = I I K K for finite extensions K/L. We fix an algebraic closure L of L and take a finite extension K/L inside L which contains all the isomorphic images of the simple components of the Hecke algebra. The elements Fi corresponding to the i-th projection X[ to K in HomA(Hord(%;K),K) = Mord(x;K) give a basis of Mord(%;K) consisting of com-

220

7: A-adic forms, Rankin products and Galois representations

mon eigenforms of all Hecke operators T(n) whose coefficients at n are given by A,i(T(n))e K. Thus we have Theorem 6. For a finite extension K in L, Mord(%;K) and Sord(%;K) have basis consisting of common eigenforms of all Hecke operators. If one normalizes such a basis of Sord(%;K) so that coefficients of q are equal to 1, then the basis elements are contained in Sor (%;l), where I is the integral closure of A in K. The last assertion follows from the fact that Hord(%; I) is finite over A and hence

Let us explain a little about the meaning of a common eigenform F e Sord(%; I). The evaluation of power series at e(u)u k -l gives an algebra homomorphism A -> 0, whose kernel is generated by Pk,e = X-(e(u)u k -l) for a finite order character e : W/WpCt —> Q p . Since I is a A-module of finite type and is integrally closed, we can find a prime ideal P of I such that PflA = Pk,eA. We identify A/P^A with 0[e], which is a subring of Q p generated by the values of e over 0. Then I/P can be identified with a finite extension of 0. Thus there exists an algebra homomorphism (p from I into Q p extending the evaluation morphism F(X) h-> F(e(u)u k -1). We identify I/P with a finite extension of 0 by taking such a homomorphism 9 ((p may not be uniquely determined if I/P is a non-trivial extension of 0). Anyway, we see that for an element F = £°° a(n,F)q n e S ord (%;I), taking the reduction F mod P is n=l

equivalent to considering the formal power series cp(F) = Z°° cp(a(n,F))qn n=l

e

Q P [[q]]. We now show that q>(F) e 5k(r o (p a p),exco" k ;Q p ). In fact,

S o r d (x;l)®i(l/P) = (S ord (x;A)® A l)(x)|(|/P) = Sord(x;A)(8)A(l(8)l(|/P))= (S0rd(%;A)(8)AA/PkA)®0(p(l) which is isomorphic to 5krd(ro(pap),£%co~k;(p(l)). In other words, by expressing F = £ i ^ i F i with FiE Sord(%;A) and X{ e I, we see that (p(F) = Si(p(>,i)Fi(e(u)uk-l) G 5krd(r0(pap),e%co"k;(p(l)). If F is a common eigenform of all Hecke operators, then cp(F) is a common eigenform of all Hecke operators in 5krd(ro(pap),exco"k;(p(l)). Hence by Proposition 2.1, 1 and some finite order character E : W —» Q p x . Each point in ^ ( 1 ) is called an arithmetic point of

7.4. Two variable p-adic Rankin product

221

Spec(l)(Q p ). Then {F(P)j for P e #(l) gives a p-adic family of common eigenforms parametrized by Spec(l)(Q p ) = Homo_aig( I, Q p ). If I = A, then we may identify the set of integers > 2 with a subset of A( I) by k t-> P k = P ^ , and we get the p-adic eigen-family of modular forms in the sense of §1. Usually we do not need to extend scalars to a non-trivial extension I; in particular, if there is no congruence modulo p between eigenforms in 5k(ro(pap),e%co"k;Qp) for at least one pair (k,%), it is known that hord(%;A) is isomorphic to a product of copies of A. We will return to this question later. Theorem 7. Each normalized common eigenform of all Hecke operators in ^k rd (r o (p a p),exco" k ;Q) for k > l is of the form (p(F) for a normalized common eigenform F in Mord(%;l) for a suitable extension I of A. The same assertion also holds for cusp forms. Proof. Let f be a normalized common eigenform in ^4(ro(pap),e%co"k;Qp) for k > 1. By extending scalars to a finite extension of 0, we may assume that f has coefficients in 0. We already know from Theorem 3 that f lifts to a A-adic ordinary form F. Then there exists an algebra homomorphism XQ of the Hecke algebra H of M ord (x;A)/P k , £ M ord (x;A) (= ^ r d (r o (p a p),e%co- k ;0) if k > 2 ) into 0 such that Ao(T(n)) = a(n,f). By definition, we have a natural surjective algebra homomorphism Hord(%,A) -> H taking T(n) to T(n). Pulling back Xo to Hord(%,A) via the above homomorphism, we have an algebra homomorphism ^o : Hord(%,A) -» 0. Let p be a minimal prime ideal of hord(%,A) contained in Ker(X0). Then I1 = Hord(x,A)/p is a finite extension of A. Let I be the integral closure of A in the quotient field K of I'. Then I' is contained in I and A,o factors through the natural projection X: Hord(x,A) —» I. That is, there exists an algebra homomorphism cp of I into Q p such that Xo = (p°X. This shows that f = cp(F) for the common I -adic eigenform F corresponding to X.

§7.4. Two variable p-adic Rankin product In this section, we construct the two variable p-adic Rankin product which interpolates the values Z)(k-l,f,g) where f and g vary on the families {f = F(u k -l)} k and {g = G(ul-l)}l for two A-adic forms F and G. Here k is the weight of f and F is assumed to be ordinary. Thus this two variable interpolation is purely non-abelian and does not include the abelian (cyclotomic) variable. We extend this two variable p-adic L-function to a three variable one in §10.4 including a cyclotomic variable. For interpolation with respect to the cyclotomic variable, there is another method due to Panchishkin [Pa, IV].

222

7: A-adic forms, Rankin products and Galois representations

We start with an abstract argument. Let S be a space of modular forms with the action of Hecke operators T(n). We assume that S is an A-module of finite type for a noetherian integral domain A. Let h(S) be the Hecke algebra of S over A, i.e. h(S) is the subalgebra of EndA(S) generated over A by Hecke operators T(n) for all positive n. We assume (51) (52) (53)

S is embedded into A[[q]] via the q-expansion f v-> X n =o a(n»QQn» S = HomA(h(S),A) via the pairing = a(l,f | h); 1I(S)(8>AK is semi-simple for the quotient field K of A.

By semi-simplicity, we have a non-degenerate pairing ( , ) on D = 1I(S)(8>AK given by (h,g) = Tro/K(hg). By this, we have a natural isomorphism i : D —> D* given by i(h)(g) = (h,g). Thus we have i"1 : S(K) = D* -> D = S(K)*. Hence we have the dual pairing ( , )A : S(K)xS(K) —» K given by (fJg)A = i"1(f)(g)« By definition, (f,g)A satisfies (la)

(f|h,g) A = (f,glh) A .

We call this pairing (, )A the algebraic Peters son inner product. In particular, if f I h = X(h)f for an algebra homomorphism X : h(S) —» A with a(l,f) = 1 (i.e. f is the normalized eigenform), the number K is well defined. In fact, by (S3), after extending the scalar field K by a finite extension if necessary, S(K) has a basis consisting of normalized eigenforms. Then c(f,g) is the coefficient of f when we express g as a linear combination of normalized eigenforms. Now we return to concrete examples. First we consider 5k(ro(pa),%o)- On this space, we have a Petersson inner product ( , ) . As seen in (5.3.10b), if we modify ( , ) to define a new product (,)«, by

for x = L «

OJ>

then we see that (f I h,g)oo = (f,g I h)*, and hence we again have, by (5.5.3) c(f g)

'

" (f,f) c "

where (, )c is as in (la) for A = C. Now we suppose that

7.4. Two variable p-adic Rankin product (PI) f = I ~

223

?Lo(T(n))qn and h = I°° (po(T(n))qn for two algebra homomor-

n=l

n=l

phisms with k > /,

U : h°krd(ro(pa),xo;Q(xo)) -> Q and (po: hKro(pp),vo;Q(vo)) -» Q; (P2)

%o is either a primitive character modulo p a or the identity character modulo p (i.e. a = 1).

By the ordinarity assumption on A., the condition (P2) actually follows from (PI). Then we define (2) L(s,X

where we have written

n(

1

-l

3

qO}"

1

and

To make computation easy, we make the following assumption: (P3)

XoW 1 is primitive modulo pY with y = max(cc,P).

The condition (P3) ensures that

which is the key in the computation in §5.4. Then we replace g in (lb,c) by hEk-KYo^Xo) for he

fWKr o (p p ),Vo). Note that f \ x = (p a ) (k " 2)/2 W(X 0 c )f for

an algebraic constant W(^o) with | W(^o) 1 = 1 . Then if a > p, we have by the same computation as in §5.4, for A = Q(^o»9o)

anc

* E=

^

(3) c(f,hE) = WZ* (f,f)A where Q(^o>9o) is the field generated by the values of ^o and (po- Now let K be a finite extension of the quotient field L of A and I be the integral closure of A in K. We consider an I-adic normalized eigenform F e Sord(%o;l). We consider the I-algebra homomorphism X : h o r d ( X ; l ) -> I given by F | T(n) = X(T(n))F. Then for ) = hE(u-/X+(u-/-l);q) with E(X;q) =

224

7: A-adic forms, Rankin products and Galois representations

we have an element Lp(Xc®(po) e K (for the quotient field K of I) given by (4)

_ , . c ^ , (F,e(h*E))i Lp(?ic
Now any element L in I can be considered as a function on X(l) = Hom O -aig(l,Q P ) = Spec(l)(Qp) by L(P)=P(L). When

I = A, we have X(A) = {x e Q p | | x - l | p < l }

via X H P X with PX(O) = P I A . Note that Sord(%;K) = KFe(KF) 1 for the orthogonal complement (KF) X of KF under ( , ) | . If P e A(\) with P | A = Pk,e and if e%co'k is primitive modulo p a p , then 5k(ro(ppa),excQ"k;K[e]) = K[e]f+(K[e]f) ± for f = F(e(u)u k -1) and the orthogonal complement (K[e]f)x of K[e]f under (» )K[£]- Localizing at P, we see that Sord(X;lp)/PSord(%;lp) = 5rd(ro(ppa),e%co"k;K[e]) by Theorem 3.3. In particular, we have (, )K[E] = ( »)l m ° d P- This shows that if we write g = eChEk-KVo^Xp)) and G = e(h*E ¥ -i x ) and define Xp = X mod P by PoX which factors through 5k(ro(pa),e%co"k;0[e]) by Theorem 3.3, we have mod P as long as %0 = %p = e%co'k is primitive modulo p a with a > (3 > 0, and %o> Vo» ^o = ^p and (po satisfy the conditions (Pl-3). We compute the value Lp(Xc®(po) when p > a. We have

= XP(T(p))

JJ^

if 5 = max(p-a.O).

Since T(p) decreases the exponent of p in the level by one as long as the character of the modular form is imprimitive, taking 8 to be the difference of the expo-

7.4. Two variable p-adic Rankin product

225

nent of p in the level of hEk./Ol/o^Xp) and a, we get the last equality in the above formula. Note that, supposing (5 > a, p ° 5 Vo(p

a

)].

Thus we see that

= (hEtKW^p) I [r O (p P )Q p ° 8 ]r 0 (p a )],f I ^ p ) ^ I * I [r o (p a )[ P Q 5 °]ro(pp)])rO(PP)Since ( P ^ j]r O (p a )[ P o 5 j]nr o (p p ) = r o (p p ), we see that 0 This implies, if Xp is primitive modulo p a with a > 0, p) I T(p) 8 )~ = p 80c -°(hE k ./(W 1 Xp).(f c I ^)(P8z))r0(pP)

Thus we have, in general, if XP is primitive modulo p a (a > 0), under (Pl-3), for 8 = max(p-a,0),

Thus we get Theorem 1 (one variable interpolation). Let X : h o r d (%;l) —» I be an l-algebra homomorphism and (po • hk(ro(p^),>|/o;0) -> Q p be an O-algebra homomorphism. Then we have a unique p-adic L-function Lp(A,c(8>(po) G K with the following evaluation property: for P G J4(l) with primitive %p modulo p a (a > 0) and P I A = Pk,e for integers k > / > 0, assuming (Pl-3) for Xo = A,P a«J %0 = %p, we have = p max(a ' P)(k - /) {^ P (T(p))- 1 9o(T(p))} max(p - a ' O) r(k-/)r(k-i)L(k-i,y
where F e S ord (x;l) is the normalized eigenform attached to X. We remove the condition (P3) in §10.5. We now want to vary (po along another p-adic family. Let M be another finite extension of L and J be the integral closure of A in M . Let G be another J-adic cusp form, i.e.

226

7: A-adic forms, Rankin products and Galois representations

G = E°° n a(n;G)q n e S(\|/,J). We suppose that G is a normalized eigenform of n—u

all Hecke operators. Henceforth, for each arithmetic point P e ^t(J), we write k(P) = k and ep = e if PI A = Pk,e- We also write the conductor of e as p a(P) p. An arithmetic point P e .#(J) is called admissible (relative to G) if for some p and \|/p = e\|/co"k. If G is ordinary, all arithmetic points are admissible relative to G by Theorem 3.3. Then we define the two variable convolution product G*E(%\\fA) as follows. We consider another copy of A identified with O[[Y]]. We regard J as an O[[Y]]-algebra and take the completed tensor product A<§>oJ = 0[[X]]®oJ = J[[X]]. We define 1

Then, for the A-algebra homomorphism (Q e #(J)), we have

for E = id®Q : A ® o J —» A = O[[X]]

id®Q(G*E) = G(Q)E(eQ(u)-1u-k(Q)X+(eQ(u)"1u-k(Q)-l)) = G(Q)*E. If one has a formal q-expansion H = Z°° a(n;H)q n e A ® oJ[[q]] and if id®Q(H) = X I = 1 id(S>Q(a(n;H))qne M(X;A) for all arithmetic points Q e ^[(J) with k(Q) > a (for a given integer a > 0), then we claim that (5) H e M(%;A)<§>oJ. To see this, write H = Z = Ti^f[rx]]/L[[X]](**H). This shows that id®Q(H#) e M(%;A) for almost all admissible Q because the denominator of O* has only finitely many roots in X(J). Thus to prove (5), we may assume that J = A. Then we write H = H(X,Y;q)e O[[X,Y]][[q]]. Since P k is admissible for all sufficiently large k relative to G by definition, we write a for the integer such that P k is admissible if k > a . We put H 0 (X) = H(X,u a -l). Then H(X,Y)-H 0 (X) vanishes at u a -l, and thus Hi = (H(X,Y)-H(X,u a -l))/Yi (e 0[[X,Y]]) for Yi = Y-(u a -l) satisfies Hi(X,u k -l) e M(x;A) for all k > a+1. Now we define Y2 = Y-(u a+1 -l) and repeat the above process to get H 2 = (Hi(X,Y)-H 1 (X,u a+1 -l))/Y 2 , satisfying H 2 (X,u k -l) e M(%;A) for all k > a+2. We then define inductively

7.4. Two variable p-adic Rankin product

227

Then Hj(X,u a+j -l) G M(%;A) and H = X~ =o H n (X,u a+n -l)n]LiYj, which is convergent under the adic topology of the maximal ideal of 0[[X,Y]]. This shows ord H E M(X;A)<8>OJ. Since the projection e : M(%;A)-^ M (%;A) extends to (6)

e : M(%;A)<§>oJ -> Mord(x;A)<§>oJ,

we can think of e(G*E(%\|/~1)), which satisfies id®Q(e(G*E(x V " 1 ))) = e(G(Q)*E0t\|T1)). Let h(\j/;J) be the J-subalgebra of Endj(S(\j/;J)) generated by Hecke operators T(n) for all n and 9 : h(\|/;J) - > J be the J-algebra homomorphism given by G I T(n) = (p(T(n))G. For each admissible point Q e .#(J),

Q p for \j/ Q = eQ\|/co-k(Q) attached to G(Q). Now considering e(G*E%X|ri) G M o r d (x;l)®oJ and extending the scalar product (,)i on Mord(%;l) to ( , ) I * J : M o r d ( x ; l ) ® o J x M o r d ( x ; l ) ® o J -> H 1 (for the denominator H of (, )i) by J-linearity, we define Lp(?iC(p) =

^py^

Then, after specializing Lp(?tc®(p) along id(E)Q, we get the function Lp(A,c®(pQ) in Theorem 1. Thus we have, regarding Lp(kc®§) as a function on X(l)xx(J) by Lp(?Lc((>)(P,Q) = P®Q(Lp(?ic®(p)), Theorem 2 (Two variable interpolation). Let X : h o r d ( x ; l ) -> I be an I-algebra homomorphism and G be a normalized eigenform in S(\|/;J). For each admissible point Q of A(J), we write (pQ : hk(Q)(ro(p^),eQ\(/co'k(Q);0[£]) -> Q p for the O[e]-algebra homomorphism associated with G(Q). Then we have a unique ip-adic L-function Lp(?ic®(p) in the quotient field of ld>oJ with the following evaluation property: for P e JZ(l) with primitive %? modulo p a (a > 0), and admissible Q e ,#(J) with \}/Q modulo p^, assuming (PI-3) for Xo = Xp, (po = 9Q and %o = Xp* Vo = VQ> we have

r(k(P)-i)r(k(P)-k(Q))L(k(P)-u P c ®(Po) G(Zp-VQ)(-2 7t V=I) k ( p «Q)(47 t ) k ( p )- 1 (F(P),F(P)) ro(p a )

*

K W

where F G Sord(%;l) is the normalized eigenform attached to X.

K

^'

228

7: A-adic forms, Rankin products and Galois representations

Note that XE(id)(X) I x=o = ^(l—)/og(u). Therefore, if \|/ = %, we have e(G*XE(id))(X,X) = 5(l-i)tog(u)e(G). Noting that, by definition, (F,F)| = 1, we have Theorem 3 (residue formula). Under the notation of Theorem 2, suppose that % = \|/. Then we have

§7.5. Ordinary Galois representations into GL2(Z P [[X]]) Now we shall explain Wiles' method [Wil] of constructing Galois representations attached to each I -adic common eigenform F. Let K be the quotient field of I. A Galois representation n : Gal(Q/Q) -> GL2(K) is said to be continuous if there exists an l-submodule L of K2 such that L is of finite type over I, L®|K = K2, L is stable under n, and as a map n : Gal(Q/Q) -> End|(L) is continuous under the m-adic topology on End|(L) for the maximal ideal m of I. Since L is of finite type over I, there is a surjective homomorphism of I-modules

L. Thus if we write m for the maximal ideal of I, then L/IT^L is the surjective image of (l/m1)11 which is a finite module. This shows that End|(L) is a profmite ring and hence compact. Thus it is natural to consider continuous representations into End|(L) from the absolute Galois group which is compact under the Krull topology. On the other hand, I is a huge ring of Krull dimension 2, and thus K cannot be a locally compact field [Bourl, VI.9.3]. This implies that the image of a continuous representation of Gal(Q/Q) into GL2(K) under any topology which makes K a topological field is very small. This is the reason why we take the m-adic topology on End|(L) to define the continuity of K. This definition of continuity does not depend on the choice of L by the Artin-Rees lemma [Bourl, IH.3.1]. We also say that K is unramified at a rational prime q if the kernel of 71 contains the inertia group of q (§1.3). We first state the result: Theorem 1. Let F be an I-adic normalized eigenform in Sord(%;l) corresponding to the A-algebra homomorphism X : hord(%,A) —» I. Let K be the quotient field of I. Then there exists a unique Galois representation n : Gal(Q/Q) -> GL2(K) such that (i) n is continuous and is absolutely irreducible, (ii) 71 is unramified outside p, (Hi) for each rational prime q prime to p, det(l-7c(Frobq)T) = 1

2

7.5. Ordinary Galois representations into GL2(Zp[[X]])

229

where Frob q is the Frobenius element at q and K : W = l+pZ p —» A x is the character given by K(US) = (1+X)S and (x) = ©(x)'^ G W. This theorem was first proven in [H4], but here we give a different construction found by Wiles [Wil]. Before giving a proof of this fact, let us explain a little about the reduction of the representation modulo a prime ideal of I (or modulo each point of 5i(\)). Let P be a prime ideal of I. We sometimes identify P with the algebra homomorphism P : I —> I/P given by reduction modulo P. For each element XG I, we regard X as a function on Spec(l) via X(P) = P(X) = X modP G I/P. We want to reduce K modulo P; thus, we consider the representation of Gal(Q/Q) on L/PL. It should look like a representation n' into GL2(Kp) (Kp is the quotient field of I/P) such that: (la) 7i' is unramified outside p; (lb) det(l-7c'(Frobq)T) = l-^(T(q))(P)T+x(q)K«q»q- 1 (P)T 2 for all prime q outside p. If PI A = Pk,eA, then A,(T(q))(P) is equal to a(q,F(P)) and

A Galois representation 7i(P) into GL2(Kp) for an algebraic closure Kp of Kp is called a residual representation of % if 7i(P) is continuous under the m-adic topology on Kp, semi-simple and satisfies the conditions (la,b). Since I is of Krull dimension 2, Kp is always locally compact under ITl-adic topology for P & {0}, and thus the continuity of K modP is clear. Since L may not be free of rank 2 over I, it is not a priori clear that the residual representation exists for all prime ideal P. In fact it exists: Corollary 1. For every prime ideal P, the residual representation TE(P) of % exists and is unique up to isomorphisms over Kp. Proof. We proceed by induction on the height of P. Suppose that P is of height 1. Since A is a unique factorization domain [Bourl, VII], the localization Ap of A at any PflA is a valuation ring. Then, the localization A at P is a finite normal extension of Ap. Therefore A is also a valuation ring. We take an l-submodule L of K2 of finite type which is stable under n and L®K = K2. Consider V = L® | A, which is stable under n and V®K = K2. Therefore V is a free A-module of rank 2. Identifying V with A2, we have a Galois representation n into GL2(A) satisfying the conditions of Theorem 1. Then by reducing n modulo P and taking its semi-simplification, we obtain a residual Galois representation 7i(P). Uniqueness of % follows from the fact that Tr(7c(P)(Frobq)) is given by A,(T(q))(P) and that Frobq is dense in the Galois

230

7: A-adic forms, Rankin products and Galois representations

group of the maximal unramified extension outside p of Q. Now we replace A by the normalization of A/PflA and I by the integral closure I' of A/PflA in the quotient field of I/P. For any prime ideal P' of height 1 in I1, we apply the same argument to K mod P and get the residual representation (n mod P) mod P . This representation is just the residual representation attached to a prime ideal in I of height 2, which is the pullback of P' in I. Continuing this process, we get the claimed result for all prime ideals of I. Even if the actual representation n is not known to exist, we can consider the residual representations separately. Thus if an I -adic common eigenform F is given, then for each point P e X(l), we call a semi-simple Galois representation 7c' into GL/2(Ap) a residual representation modulo P if n' satisfies the conditions (la,b) for P, where Ap is the integral closure of Op in its quotient field. Our method of proving the theorem is to show that the desired % exists if there exist infinitely many distinct primes {P} in X(l) which have the residual representation modulo P. Such a family of infinitely many residual representations is supplied by the following theorem of Deligne: Theorem 2 (P. Deligne [D]). Let M be a finite extension of Q p . Let X : hk(ro(N),%;Z[%]) —> M be an algebra homomorphism. Then there exists a unique Galois representation n : Gal(Q/Q) -» GL2(M) such that (i) n is continuous and absolutely irreducible over M, (ii) n is unramified outside Np, (iii) for each prime q outside Np, det(l-7i(Frobbqq)X) ) = 1 - 5i(T(q))X + x(q)q k " 1 X 2 . This result in the case where k = 2 (except the determination of ramified places) is a classical result due to Eichler and Shimura and its proof can be found in [Sh, Th.7.24]. The unramifiedness outside Np was later proven by Igusa in the case of weight 2. The case k > 2 is treated in Deligne's work [D]. The remaining case k = 1 is dealt with by Deligne and Serre in [DS]. Note that %(-l) = (-l) k and thus det(rc(c)) = %(-l)(-l)k~l = -1 for complex conjugation c. For further study of ramification, see [La] and [Cl]. Then Theorem 1 follows from the above theorem of Deligne and the following result of Wiles: Theorem 3 (Wiles [Wil]). Let F be an l-adic normalized common eigenform and suppose that there exists an infinite set S of distinct points in X(\) such that for every P e S, the residual representation n(P) into GL/2(0p) exists, where Op is the p-adic integer ring of the quotient field of I/P. Then there exists a Galois representation n : Gal(Q/Q) -> GL2(K) satisfying the conditions of Theorem 1.

7.5. Ordinary Galois representations into GL2(Zp[[X]])

231

In fact, if F is ordinary, we can take A(l) as the set S by the theorem of Deligne. Even if F is not ordinary, we can take as S the set of admissible points in A(l). For ordinary F, we can avoid the use of infinitely many modular forms of different weight, taking as S the set {P e .3(1) | k(P) = k}. Thus for a fixed weight k > 1, if we have Deligne's Galois representation for every common eigenform of weight k, Theorem 1 follows from Theorem 3. In particular, if we take k = 2, Theorem 1 follows from the result in [Sh, §7.6], which shows the existence of Galois representations attached to modular forms of weight 2. Now we start the proof of Theorem 3. Although Theorem 3 holds even for p = 2, we prove the theorem assuming p > 2 for simplicity. We refer to [Wil] for the proof in the general case. Let us fix P e S and write n for TC(P) and M for Kp. Let A be the p-adic integer ring of M. Thus n has values in GL2(A). Let G be the Galois group of the maximal extension unramified outside p. Since n is unramified outside p, we may consider % as a representation of G. We write L for A2 and consider L as a G-module via K. We write c for complex conjugation. Since c 2 = 1 and det(7t(c)) = - 1 , the eigenvalues of 7t(c) are ±1. We decompose L = L+©L_ into the sum of the ±1 eigenspaces of 7t(c). Thus by identifying L with A via the basis of L±, we may assume that -1 0

For each

a € G, we write ft (a) =

x: GxG —» A by properties:

X(G,X)

f() (

,

()

,( J

and define a function

= b(a)c(x). Then these functions satisfy the following

(2a) as functions on G or G2, a, d and x are continuous, (2b) a(ox) = a(a)a(x)+x(a,x), d(ax) = d(a)d(x)+x(x,a) and x(ax,py) = a(a)a(y)x(x,p)+a(y)d(i:)x(a,p)+a(a)d(p)x(T,Y)+d(T)d(p)x(a,Y), (2c) a (l) = d(l) = d(c)= 1, a(c) = -l, and x(a,p) = x(p,x) = 0 if p = 1 or c, (2d) x(a,x)x(p,rt) = x(a,Ti)x(p,x). The properties (2c) and (2d) follow directly from the definition, and the first half of (2b) can be proven by computing directly the multiplicative formula b(o)Va(x) bCcfi _ fa(ox)

^

d(a)J[c(x) d(x)J " tc(ax) d(ox)J* Then, in addition to the two first formulas of (2b), we also have

232

7: A-adic forms, Rankin products and Galois representations

b(ax) = a(a)b(x)+b(o)d(x) and c(ax) = c(a)a(x)+d(a)c(x). Thus we know that x(ax,py) = b(ax)c(py) = (a(a)b(x)+b(a)d(x))(c(p)a(y)+d(p)c(Y)) = a(a)a(Y)x(x,p)+a(y)d(x)x(a,p)+a(a)d(p)x(x,Y)+d(x)d(p)x(a,y). For any topological algebra R, we now define a. pseudo-representation of G into R to be a triple TC' = (a, d, x) consisting of continuous functions on G or G2 satisfying the conditions (2a-d). We define the trace Tr(7c') (resp. the determinant det(Tc')) of the pseudo-representation n1 to be a function on G given by Tr(7t')(cO = a(a)+d(o) (resp. det(7c')(cj) = a(o)d(a)-x(o,a)). Our proof of Theorem 3 is divided into two parts: the steps are represented by the following two propositions: Proposition 1. Let TC' = (a, d, x) be a pseudo-representation of G into an integral domain R with quotient field Q. Then there exists a continuous representation % : G —» GL2(Q) with the same trace and determinant as K\ Proposition 2. Let a and B be two ideals of I. Let n(a) and n(B) be pseudo-representations into I/a and MB, respectively. Suppose that n(a) and K(&) are compatible; that is, there exist functions Tr and det on a dense subset E ofG with values in l/of]B such that for all O G Z , Tr(7i(a)(o)) s Tr(a) mod a and Tr(n(B)(o)) = Tr(a) mod B, det(7c(a)(a)) = det(a) mod a and det(7t(£)(a)) = det(a) mod B. Then there exists a pseudo-representation %{cf\S) of Q into \lcP\B such that Tr(n(cf]B)(c)) = Tr(a) and dtt(n(c{]B)(o)) = det(a) on E. First admitting these two propositions, let us prove Theorem 3. Since G is unramified outside p, the set E of Frobenius elements for primes outside p is dense in G (Chebotarev density theorem: Theorem 1.3.1). We put Tr(Frobq) = ?i(T(q)) and det(Frobq) = %(q)K((q))q"1. We number each element of S and write S = {Pi}?\ and K{ for 7i(Pi). We construct out of each residual representation 7Ci for P e S, a pseudo-representation n\. Then all the TCVS are compatible. Then by the above proposition, we can construct a pseudo-representa-

tion Kd into 1/nLPj so that Tr(n\c)) = TT(%*-\O))

mod flrKp\ on E.

Both sides of this congruence are continuous functions and hence Tr(7Cfi(a)) = TrCTc'1"1^)) mod Pifl ••• HPi-i on G. Note that by definition, if %{ = (ai, di5 xO, ai(a) = 2" 1 (Tr(7c' i (a))-Tr(7i' i (ac))) and

di(a) = 2" 1 (Tr(7c fi (a))+Tr(7i: ti (ac)))

and xi(a,x) = ai(ax)-ai(a)ai(x).

Therefore we have ai(a) = ai.i(a) mod Pifl ••• DPi-i, di(a) = di.i(o) mod Pifl ••• PlPi-i

7.5. Ordinary Galois representations into GL2(Zp[[X]])

233

and xi(a,T) = xi_i(a,x) mod Pifl ••• HPi-iThus we can define a pseudo-representation 7c1 into I = Km l/PiPl-'-PlPi by i a

7C'(CJ) = lim 7c (a).

Then we can construct the representation K out of tf by Proposition 1. Wenow prove Proposition 1. We divide our argument into two cases: Case 1: there exist p and y e G such that x(p,y)^0, and Case 2: X(G,X) = 0 for all a,x in G. (a(a) C Case 1. We define n(o) = \(a) by putting d ( J c(a) = x(p,a) and b(a) = x(a,y)/x(p,y). Then b(a)c(x) = x(p,x)x(a,y)/x(p,y) = x(a,x) by (2d). Thus we know that the entry of 7c(a)7c(x) at the upper left comer is equal to, by (2b), a(a)a(x)+b(a)c(x) = a(a)a(x)+x(o,x) = a(ox). Similarly the lower right comer of 7i(o)7t(x) is equal to d(G)d(x)+b(x)c(a) = d(a)d(x)+x(x,a) = d(ox). We now compute the lower left comer of 7i(a)7c(x), which is given by c(a)a(x)+d(a)c(T) = x(p,a)a(x)+d(a)x(p,x). By applying (2b) to (l,p,a,x), we have c(ax) = x(p,ax) = a(x)x(p,a)+d(a)x(p,x), since x(l,c) = x(l,x) = 0 by (2c). This shows that c(ax) = c(a)a(x)+d(a)c(x). Similarly by applying (2b) to (a,x,l,y), we have b(ax)x(p,y) = x(ax,y) = a(a)x(x,y)+d(x)x(a,y) = (a(o)b(x)+d(x)b(a))x(p,y), which finishes the proof of the formula 7t(o)7i;(x) = 7c(ax). Obviously, by defini(l 0\ tion, TC(1) = Q and hence TC is the desired representation. Case 2. In this case, by (2b), we have a(a)a(x) = a(ax) and d(a)d(x) = d(ax) for all c,XEi G. Then we simply put n(o) =

fa(o) ^

0^

., J which does the job.

234

7: A-adic forms, Rankin products and Galois representations

We now prove Proposition 2. We consider the exact sequence: 0 -> l/aPi5-> \la®M6 —2-> \/(a+S) -> 0 ai-> a m o d a © amod£ a©b h-» a-b mod a+B. We consider the pseudo-representation n = n(a)®K(6) with values in l/a®l/5. The function a°Tr(7t) vanishes identically on £. Since this function is continuous on G and £ is dense in G, a°Tr(7c) vanishes on G. Thus Tr(rc) has values in l/dT\b. If we write n = (a, d, x), then a(Q) = 2-1(Tr(7c(a))-Tr(7i(ac))), d(a) = 2"1(Tr(7c(a))+Tr(7c(oc))) and x(o,x) = a(ox)-a(a)a(x). Thus n itself has values in l/cP\5 and gives the desired pseudo-representation.

§7.6. Examples of A-adic forms In this section, we briefly discuss some examples of ordinary and non-ordinary A-adic cusp forms. We will not give detailed proofs but satisfy ourselves with indicating the source where one can find proofs. We start with the lowest weight cusp form of SL2(Z), which is the Ramanujan A function. The function A spans 5i2(SL2(Z)) and is a normalized eigenform. Then it is known by computation that A | T(p) = x(p)A with x(p) e Z p TlZ for 11 < p < 1021. Thus writing as a the unique p-adic unit root of X 2 -x(p)+p 11 = 0 for those primes, we know that f = A(z)-a"1p11A(pz) is a normalized eigenform of level p and f | T(p) = ocf, i.e. f is ordinary. Then, if | x(p) | p = 1, the fact that 5iO2d(r0(p);Qp) = Qpf follows from [M, Th.4.6.17 (2)]. In general, we have diir^ rd (r o (p);Q p ) = dimj£rd(SL2(Z);Qp) (Proposition 7.2.2). Thus h° rd (r o (p);Z p ) = Z p via T(n) H> x(n) which is the eigenvalue of T(n) for A. This implies hord(co12;A)/Pi2hord(co12;A) = Z p . Since hord(co12;A) is a A-algebra, we have the structural morphism i : A —> hord(co ;A). Then by the Nakayama lemma, we know that i is surjective. As we have already seen (Theorem 3.5), hord(co12;A) is A-free, and hence i is an isomorphism. Thus there exists a unique ordinary A-adic normalized eigenform FA spanning Sord(co12;A) such that F A (u 12 -l) = A(z)-a" x p n A(pz) as long as I x(p) | p = 1 holds. Since 12 is the least weight k for which 5k(SL2(Z)) * 0, h ord (x;A) = 0 for p = 2, 3, 5 and 7 for all %. In the text, we have only discussed A-adic forms of level p°°. If we introduce an auxiliary level N prime to p, there are abundant examples. A formal q-expansion F G A[[q]] is called a A-adic form of level Np°° and of character % for a finite

7.6. Examples of A-adic forms

235

order character % of (Z/NpZ) x , if for almost all positive integers k, F(uk-1) G fA4(ro(Npa),%co"k;0) for some fixed a. Then we define the notions of A-adic cusp forms, A-adic ordinary forms, etc. in the same manner as in §7.1. The first example of this type is the A-adic cusp forms associated with imaginary quadratic fields (given by theta series). We thus fix an imaginary quadratic extension M/Q. Then there are abundant arithmetic Hecke characters X such that M(&)) = oc^1 for a = 1 mod c for some ideal c, where k > 1 is a positive integer. The largest ideal c in the integer ring r of M with this property is called the conductor of X. Then it is well known (see [M, Th.4.8.2]) that there exists a modular form

(1)

h = 5>(*)q" (fl)

where D is the discriminant of M/Q, %(m) = f — ] is the Jacobi symbol and X(m) = X((m))/rr^~1 for integers m. This form is known to be a cusp form if X is non-trivial (in particular if k > 1). Then fx, is a normalized eigenform and for primes /, a(/,f) = 0 if Ir is a prime ideal, a(/,f) = X(C)+X(C) if lr= CC with I and CC + c = r, a(/,f) = X(C) if Ci) Dc. Here if /"=> c, we agree to put i = 0. Let p be a prime ideal of M given by {xe r\ | x | p < l } for the p-adic absolute value of Q p . We fix one character X modulo cp for an ideal c prime to p such that X((a)) = a if a = 1 mod cp. Then we take K = QP(X) and its p-adic integer ring O. We decompose (7* = WKX|LIK SO that WK is Zp-free and JJ^K is a finite group. We then write the projection map of cf onto W K as X H (X). First we suppose that pr= pp with two distinct prime ideals p and p. Then the subgroup WM of WK topologically generated by (X(a)) for all ideals a prime to p is isomorphic to the additive group Z p and the index p^ = ( W M : W ) < °°. Here we consider W = l+pZ p as a subgroup of WM by first regarding W as a subgroup of rf and then applying the map: z h-> (X(z)). Note that this is in fact the natural inclusion map of W into Qp(?i). The exponent y = 0 if the class number of M is prime to p. We fix a generator w of WM SO that wpY = u. Then we consider the ring I = O[[Y]] containing the ring A = O[[X]] with the relation (1+Y) p7 = (1+X). Then we consider the series (2a)

Fx(Y;q) = ^ a Ma)(X(a))- 2 (l+Y) s(a) q N(a) if P^= pp

with

p*p,

where a runs over all ideals prime to p and s(a) = log((X(a)))/log(w), i.e. s(fl) = (X(a)). Then we know that (1+Y)s(fl) | Y=wk-i = wks(fl)= (X(a))k. Noting w

236

7: A-adic forms, Rankin products and Galois representations

that the Hecke character ^ k modulo cp given by X^(a) = X(a)(X(a))k'2 has in fact values in Q, we get (2b)

Fx(w k -l;q) = f^k e

5k(r0(DN(c)p),%X(x)2~k;Oy

Here it is easy to compute Xk = % X(O2'k. If we substitute £w k -l (instead of w k -l) for X in F^(X;q) for a py-th root of unity £, we have (2c)

F^(Cw k -l;q) = f£^k e 5k(r0(DN(c)p),x?lC02-k;O),

where 8 is the finite order character of the ideal class group C1M(1) given by

e(a) = Cs(a)Since P k = X-(u k -l) = II;(Y-(^w k -l)) in I, we know that F^ is an I-adic form of level DN(c) and of character %X(O2. Since eXy&p) is a p-adic unit and since the eigenvalue of T(p) for fe^k is given by eX k (p), F^ is ordinary. If one chooses X so that X\ is trivial (this is always possible), then we see that

Thus FJL(W-I) =E(%CO)(U-1) where E(%co)(X;q) is the A-adic Eisenstein series such that E(%co)(uk-l)=Ek(%co1"k) or Ek(x)(z)-Ek(%)(pz) according as CO1"1" is non-trivial or not. Then, for example, supposing WM = W, we have a strange A-adic form E(X;q) = ^ ( u - n ^

This implies

rj I T , x a(n,E(%co))E(xco)-a(n,FQF^ a(n,E(%co))-a(n,F^ / E|T(n)= x ^ ^ = x^j) E(Xco)+a(n,FOE. Note that — l —T?—,—y— 2 —E A

and is non-vanishing at u-1. Thus on

Mord(%co;A)/PiMord(%co;A), T(n) acts non-semi-simply. On the other hand, the action of Hecke operators T(n) for n prime to Dp on f^i(ro(Dp),%co) is semi-simple, because the same argument as in the proof of Theorem 5.3.2 obviously works. This shows that inside 0[[q]], the image of specialization of Mord(%co;A) under F H F(u-l) is larger than the space of classical ordinary forms fA^rd(r0(Dp),%co;0). Now we assume that p is inert or ramified in M. For simplicity we assume that p * 2. We have a surjective group homomorphism q>: C1M(^P°°) -> WM given by cp(x) = (X(x)). On C1M(^P°°) J there is the natural action of complex conjugation c which leaves the maximal torsion subgroup of C1M(^P°°) stable,

7.6. Examples of A-adic forms

237

and hence c acts on the image WM by (p(x)c = (p(xc). This action may not coincide with the usual Galois action of c on Q(^). Then WM = Z p 2 and thus we have two generators wi and W2 of WM. We then may assume that wi = u and W2 = w with wc = w"1 for complex conjugation c. We then write (X(a)) = us(a)wt(fl) and define (3a) Fx(X,Y;q) =

^X

This is not really a A-adic form but, so to speak, an O[[X,Y]]-adic form. That is, we have, for a pair (£,£') of p-power roots of unity (3b)

F ^ u k - l , i > k - l ) = feXk

where e(a) = ^ ^ ' ^ is a finite order character with conductor pl\ Since a(p,fx,k) is either 0 or a p-adic non-unit, the family of modular forms {fex,k} will never be ordinary. These A-adic forms constructed out of imaginary quadratic fields are called A-adic forms with complex multiplication or of CM type. As is clear from the above examples, there exist l-adic forms having values in a non-trivial extension I of A. There are examples of A-adic forms without complex multiplication. We start from a primitive finite order character ^i modulo c of the imaginary quadratic field or real quadratic field M. We continue to write r for the integer ring of M. Then, even in the case of real M, if X\{oC) = -1 for a e r with the congruence a = 1 mod c and oca = -1 mod c for a non-trivial automorphism a of M, f\ as in (1) is a cusp form in 5i(Fo(DiV(c)), X%) for the discriminant D of M and the Jacobi symbol %(m) = f — ]([M, Th.4.8.3]). Suppose that U ) (4a) (4b)

p is a divisor of DN(c) (resp. D) if M is real (resp. complex); There exists a prime factor p of p in r such that f is prime to c,

where a = c when M is imaginary. Then we see that fx\ I T(p) = X(pc)fxi and hence f^ is ordinary because X(pG) is a root of unity. When p | D, there are no ordinary A-adic forms with complex multiplication. When M is real quadratic, the modular forms associated with real quadratic fields do not exist in weight higher than 1. However by Theorem 7.3.7, we can lift f?^ to an l-adic normalized eigenform F which is not of CM type. We know from [M, Th.4.2.11 and Th.4.2.7] that

238

7: A-adic forms, Rankin products and Galois representations

(5)

dimc(5 2 (r 0 (4))) = 0 and for odd p, dimc(5 2 (r 0 (p))) = ^

It is also known that T(p) is invertible on S2(^o(v)) a n d has only eigenvalues ±1 ([M, Th.4.6.17 (2)]). Thus in this special case, we have 5 2 (r 0 (p);O) = 5| rd (r 0 (p);O). Then by Theorem 3.3, we know Theorem 1. For odd primes p, we have rank A (h ord (co 2 ;A))=rank A (S ord (co 2 ;A)) = The above dimension formula is due to Dwork when p = 1 mod 12 [K4, p. 140]. By the above formula rankA(hord(co2;A)) grows linearly as the prime p grows. This is the only case where we know the exact rank of the space of ordinary A-adic forms in a systematic way. There are several numerical examples computed by Maeda for A-adic forms. We refer to [Md] and [H2] for these examples. In particular, in [Md] Maeda gives a criterion in terms of generalized Bernoulli numbers for having non-trivial A-adic forms without complex multiplication which are congruent to A-adic modular forms with complex multiplication. Such a congruence is very important in studying the Iwasawa theory of imaginary quadratic fields and CM fields. We refer to [HT1-3] for such applications in Iwasawa's theory.

Chapter 8. Functional equations of Hecke L-functions In this chapter, after giving a brief summary of the notion of adeles of number fields, we prove the functional equations of Hecke L-functions. For further study of adeles and class field theory, we recommend [Wl] and [N],

§8.1. Adelic interpretation of algebraic number theory Let us start with the explanation of the adele ring of Q denoted by A. To define the topological ring A, we consider the module Q/Z and its ring End(Q/Z) of additive endomorphisms. Let Q/Z[p°°] be the subgroup of Q/Z consisting of elements killed by some power of p for a prime p of Z. Then by definition, for any two distinct primes p and q, Q/Z[p oo ]nQ/Z[q o °] = {0} and hence ©pQ/Z[p°°] c Q/Z. Let us show that this inclusion is in fact surjective. For any given rational number r, we expand r into the standard p-adic expansion S°°_ c n p n defined in §1.3 and put [r]p = ZnL.mcnpn (the p-fraction part). Since r

-Mp e ZpHQ, p does not appear in the denominator of r-[r]p. Repeating this process of taking out the p-fraction part from r for all prime factors of the denominator of r, we know that r-E p [r] p e Z and hence r = X p [r] p in Q/Z. Since [r] p e Q/Z[p°°], we know that ©PQ/Z[p°°] = Q/Z. The above process can be applied to p-adic numbers r e Q p . Then r — i > [r] p plainly induces an isomorphism: Q p /Z p = Q/Z[p°°]. Thus identifying Qp/Zp with Q/Z[p°°], we have (1) Q/Z s © P Q P /Z p . This in particular shows that End(Q/Z) = npEnd(Qp/Zp). The multiplication by elements of Z p induces a morphism i : Z p —> End(Q p /Z p ), i.e., i p (z)(z') = zz'. We now show that i is a surjective isomorphism. Obviously it is an injection. Note that the submodule Qp/Zp[pT] killed by p r is isomorphic to Z/prZ via x — i > prx. Since the endomorphism of Z/prZ is determined by its value at 1, we see that End(Q p /Z p [p r ]) = Z/p r Z via (p h-> p(q>) = pr(p(p~r)- If (pr e End(Qp/Zp[pr]) is the restriction of (p E End(Qp/Zp), then {p((pr)}r is coherent so that p((pr) = p((ps) mod p s if r > s. Then the map p : End(Q p /Z p ) -* Z p given by p((p) = lim p((pr) e Z p gives the inverse of i. This shows that m

(2a)

End(Q p /Z p ) = Z p and End(Q/Z) = Z =

For any finite subgroup X of Q/Z, we put A(X) = {a G End(Q/Z) | aX = 0}.

240

8: Functional equations of Hecke L-functions

Then obviously End(Q/Z)/A(X) = End(X). Declaring {A(X)}X to be a fundamental system of neighborhoods of 0 in End(Q/Z), we give the structure of a topological ring on End(Q/Z). This is the weakest topology that the quotient topology of End(Q/Z)/A(X) = End(X) is discrete, and hence it is a topology of the projective limit End(Q/Z) = lim End(X). Thus End(Q/Z) is a compact ring, which is isomorphic to Z = n p Z p as topological ring. We define the finite part Af of the adele ring A to be the subring of IIpQp generated by Q and Z. Here Q is embedded diagonally into IIpQp by Q ^ a n (•••a,a,a,---) e IlpQp. We give the structure of a locally compact ring on Af so that Z is an open compact subgroup of Af and the topology of Af induces the topology on Z. We define the adele ring A to be AfxR and give the product topology on it. Then A is a locally compact ring. If x e IlpQp satisfies xp e Z p for almost all p (i.e. for all but finitely many p), we can cancel the denominator of x by multiplying a rational integer m, i.e., mx e IIpZp. Thus x e Af. It is obvious that x p G Z p for almost all p if x e Af. Then we know that

(2b)

A f = n ' p Q p = ( x e n P Q p i XP e

Z

P f°r aim°st

an

p}-

Thus Af is the restricted direct product of Q p with respect to Z p . This is the definition of Af usually found in the literature. Now we shall see that Af = Z+Q. For any x e Af, the above definition using the restricted direct product shows that [x] = Z p [x p ] p is actually a finite sum and is a rational fraction. Then x-[x] e Z showing A f = Z + Q . In particular, we have (2c)

A f = Z+Q and Af/Z = Q/(ZDQ) = Q/Z.

The multiplicative group A x is called the idele group of Q. Since Z is a principal ideal domain, for any x e AfX, we can write xp = u p 6 ^ with u E Z p x . Since primes p with e(p) ^ 0 are finitely many, {x} = TIpp 6 ^ is a positive rational number and hence x(x)" 1 e Z x , which shows (3a)

A x = ZXQXR+X,

where R+ x = {x E R | x > 0}. Since Q c A via Q 3 a H ( a , a , . . . , a ) G A, A is naturally a Q-algebra. Consider a small open interval Ue = (-8,8) for e > 0 in R. Then O = Z x U e is an open neighborhood of 0 in A. We see that QflO = ZflU e = {0} if 0 < 8 < 1. Thus we have found an open neighborhood O of 0 in A such that QflO = {0}. Note that A/Q = {(ZxR)+Q}/Q = ( Z 0 R ) / { ( Z 0 R ) n Q } = ( Z 0 R ) / Z = Zx(R/Z), which is a compact set. Thus

8.1. Adelic interpretation of algebraic number theory (3b)

241

Q is a discrete subring of A and A/Q is compact.

For a number field F (i.e. a finite extension F/Q), we simply put FA = F ® Q A . Then we have the ring embedding F s a i - ) a® 1 G FA and hence FA is an F-algebra. The trace map Trp/Q : F -> Q induces the A-linear map Trp/Q®id : FA —> A, which is again denoted by Trp/Q and is called the (adelic) trace map. Fixing a basis {coi} of F over Q, we identify F = QtF:™ a s a vector space, which induces an isomorphism FA = A^F:(^ of A-modules. This identification gives a natural topology on FA, under which FA is a locally compact topological ring. Obviously the topology of FA does not depend on the choice of the basis. In particular, (3c) F is a discrete subring of FA and FA/F = dxF^/O is compact, where we put 0 for 0®zZ for the integer ring 0 of F. Since A = AfXR, we have FA = FAfxFoo with FA £ = F®QAf and Foo = F ® Q R . Thus we can write x = (xf,Xoo) with the finite part Xf e FAf and the infinite part Xoo e F^. Let I be the set of all field embeddings of F into C. Then Aut(C) acts on I from the right via natural composition. Then the set of archimedean places a is defined to be the set I/(c), where (c) is the group of order 2 generated by complex conjugation c. Each c e a gives rise to a complex absolute value | | a on F by I a | a = | a a | for any representative a e I. Then writing I(R) for the subset of I consisting of real embeddings and I(C) = I-I(R), i.e., the set of embeddings whose image is not contained in R. We put a(R) = I(R) and a(C) = I(C)/. Then as seen in (1.1.5b), F^ = R a ( R ) x C a ( C ) . Similarly O = UpOp, where p runs over all prime ideals of 0. The unit group FA X of FA is called the idele group of F, whose elements are called ideles of F. We give the topology on FA X SO that the system of neighborhoods of 1 is given as follows. Let 6x(m) be the kernel of the natural map YlpOpx -> (O/m)x for all ideals m of 0. Then a fundamental system of neighborhoods is given by {0 x (m)xU | wan ideal of 0, U a neighborhood of 1 in F ^ } . Thus FA X is a locally compact group. The elements of F A X are called ideles. The topology of FA X is stronger than the one induced from the adele ring FA- In fact, the neighborhood of 0 of FA is given by 0(m)xV for the kernel 6{m) of the natural map of additive group HpOp —> (O/m) and a neighborhood V of 0 in Foe. One sees easily that {l+0(m)xV}nFA X 3 0 x (w)x(l+V) but they are never equal. Exercise 1. Show that {l+0(m)xV}nF A x ^ 0 x (m)x(l+V).

242

8: Functional equations of Hecke L-functions

In the same manner as in (3a), we see that F A f x = {x e UpFp\ xp e Op for almost all p}.

(4a)

From this, writing xp0p = pe^p\ we know that e(p) = 0 for almost all p. Thus the ideal xO = Uppe^ makes sense as a fractional ideal of F. Thus we have a natural homomorphism of groups FAfx -> /

(4b)

given by

X H XO.

Obviously this map is surjective. By abusing symbols, we write xO for XfO when x € FA X . We now define the adele norm | x | A by | Xf | Afx I Xoo I «o for I x I Af = HPI xp I p and I Xoo I co = IToe a(R) I x a I oxHoe a(C) I x o I a2- We normalize 1®P I p = N(p)'1 for the prime element U5p of Op. Then for a e F x , we see that | a | o o = l l a e a(R)

I a a I oxIIacEa
On the other hand, writing the prime factorization of aO as aO = Hpp^p\ we see from (1.2.2b) that

I a I Af = TlpN(pei*>yl

= TV(aO)"1 = | W(a) I -1.

By this we know the following product formula: I a | A = 1 for all a e F x .

(5)

Put F ^ = {x e FA X I x I A = 1}- Then by the product formula, we see that

F(i) A

-x -px —-> JP •

Theorem 1. F x is a discrete subgroup of F ^ and F ^ / F x is a compact group, where the system of neighborhoods of 1 of the quotient group F ^ / F x is given by the images of the neighborhoods of I in F ^ \ Proof. The set of normalized absolute values of F, i.e.

{I \p I I o I P: prime ideals, o e a}, is called the set of places of F. When we do not care much about the difference of finite or infinite places, we just write v to indicate a place of F. When we write p (resp. a), it indicates a finite (resp. archimedean) place. Let a be an idele and define V(a) = ( x e Ap

I x v | v < | av I v for all v}.

8.1. Adelic interpretation of algebraic number theory

243

Then we claim that the following assertion is equivalent to Corollary 1.2.1: (6)

for any idele with sufficiently large I a | A, V(a)riFx * 0 .

To see this, let a - aO. Since a = {x e F a = V(a)FcoriF. Thus we only need to show that {xea|

|xa| < |aa|

I x | v < | a v | v } , we see

for all a e a} * {0}.

Since a(V(a)nF x ) = V(aa)nF x and I a | A = I oca | A for a e F x , we may assume that a is an integral ideal. Then by Corollary 1.2.1, there exists a constant C > 0 independent of a such that if | a<x> I <« = Ilaei I a a | > CN(d), then there exists 0 ^ a E a (<=> | (Xf I A ^ I &f I A) such that I oca I < I a a | (i.e. | a a | a < | a o l a for all a). Since N(a) = UfL" 1 , U | A ^ C if and only if I aoo I A ^ C I af I A"1 = CiV(a). This shows (6). Now we shall prove the theorem. We already know that F is discrete in FA (3C). Since FA X has a stronger topology than FA, the induced topology on F x from FA X is stronger than the topology induced from FA- Since the discrete topology is the strongest, F x is discrete in FA X . Take an idele c such that | c | A ^ C. Then for arbitrary a E F ^ \ | ca"11 A ^ C. Thus we can find by (6) an element a E F x such that a E a' 1 cV(l) = V(a"1c). That is, a a e cV(l). Since a a is again in F ^ \ applying the above argument to (aa)"1, we can find p e F x such that P E aacV(l) or p(aa)" 1 € cV(l). Thus p = p(aa)" x aa e cV(l)cV(l) which is contained in c 2 V(l). Since c 2 V(l) is compact and F x is discrete, c 2 V(l)f|F x is compact and discrete and is therefore finite. Writing c 2 V(l)f|F x = {pi,..., p m }, we see that (aa)"1 e U^Pi^cVQ) for any a e F(^}. Now we put B= Then aa G B and (aa)"1 e B. Since B is a compact subset of FA, it is easy to see that B* = {x E B | X"1 G B} is a compact set of FA X . In fact, for each v, I x | v for x E B* is bounded from above and from below, i.e. 3MV > 0 such that | xv | v < M v and U U ^ M / 1 because x and x"1 e B. These Mv can be taken to be 1 for almost all v. Thus, we see that F X B* 3 F(^} and B* is a compact set of FAX. Thus F^ } /F x is compact. Corollary 1. Let I be the group of all fractional ideals of F and & be the subgroup ofprincipal ideals. Then we have

244

8: Functional equations of Hecke L-functions

which is a finite group. Proof. To any idele a, we have associated an ideal aO = aOflF. This correspondence induces a surjective homomorphism of groups p : FA X —> /. We see easily that p " 1 ^ ) = F x 0 x F x .

This shows the first isomorphism.

Since

F A X = F^F X <3 X F* by definition, we have the second isomorphism. Since F x 0 x F x is an open subgroup of FAX , the middle term is a discrete group. Since F ^ is a compact group, the last term is a compact group. Thus I/(P is a discrete compact group and hence is finite. This gives another proof of the finiteness of the ideal classes. Corollary 2 (Dirichlet-Hasse). Let S be a finite set of places. Let U ( S ) = { X E F A X | | x | v = l for all v g S} and O(S)X = U(S)OFX. Then if S contains a, i.e. S contains all infinite places, then O(S)X = JI(F)XZ S " 1 for s = #(S), where JI(F) is the group of roots of unity in F, and Z is considered as an additive group. Note that 0(Soo)x = 0* and hence the above corollary includes Dirichlet's theorem about units as a special case. Proof. Consider the group homomorphism 0 : FAX s a

H>

(log I x | v ) v e s e Z s " r x R r ,

where r is the number of infinite places and we have normalized log at each place so that the above map has values in Z at the finite places. Then Ker(6) = U(0). Thus U(S)/U(0) = Z s " r xR r . Since for the set of infinite places Soo, F ^ = F A X , we see that U(S)/F^nU(S) = FAX/F(A1} = R . Thus F ( ^nU(S)/U(0) = Zs-r x R1"1. On the other hand, F ( ^nU(S)/O(S) x U(0) = F^nF x U(S)/F x U(0) is compact but F x U(S)/F x U(0) = C<S)x/cX0)x is discrete. Thus F^nU(S)/O(S) x U(0) = (R/Z) r4 xG

8.2. Hecke characters as continuous idele characters for a finite group G and O(S) X /0(0) X = ZsA. O(0) x = JLL(F). This shows the result.

245

By Lemma 1.2.3, we see that

§8.2. Hecke characters as continuous idele characters In this section, we show that each ideal Hecke character X uniquely induces a continuous character of FA X /F X such that ^(x) = X(xO) if xc = 1 for the conductor c of X. Here xc = (xp)p^c. Let F be a number field and I be the set of all embeddings of F into Q. Let O be the integer ring of F and m be a non-trivial ideal of O. Write I(m) for the group of all fractional ideals of F prime to nu Let F v be the completion of F at v and Ov (resp. n^) be the closure of O (resp. m) in F v . (We use the symbol /v to indicate the prime ideal of O corresponding to v and also its closure in Ov if v is a finite place). We define Uv(m) = {a G Ov I a = 1 mod m v }. Thus if mv = pve, then U v W = { a e Ov

I a-1 | v < N(p)'e], a closed disk of radius N(p)'e centered

at 1, and \Jw(m) = OyX if pv is prime to nu We define U(w) = J[ J^v finite U v (m) and Foo Foo+ = {x = (x v ) G F ^ I xv > 0 if v is real}, P(m) = F x H{x G FA X xp G XJv(m) if p I m and Xoo G FOO+} = {a G F x | a G Uv(m) if py divides m and a a > 0 if a is real}. Here the intersection is taken in the idele group FA X . Let T(m) be the subgroup in I(m) consisting of principal ideals spanned by elements a G P(m). Then the ray class group Cl(m) modulo m is defined by the following exact sequence: 1 -> I(m) -> C\(m) - » 1.

We have another exact sequence 1 -> E(w)

-> P(w)

-» !P(OT) ->

1,

where E(m) = FxnU(m)Foo+ = {a G Ox I a G Uv(w) if pv divides w and cc° > 0 if a is real}. Let x = (xv) G FA X be an idele such that xv = 1 if either v is infinite or mv ^ Ov (i.e. pv divides m). Then x v 0 v = pve^ and e(v) = e(xv) = 0 for almost all v including those v appearing in nu Then we defined a fractional ideal xO in F by xO= IIv/v • This gives a group homomorphism of

246

8: Functional equations of Hecke L-functions FA(W) = ( x e FA X I xv = 1 if either v is infinite or mv * Ow]

into I(m). Lemma 1. We have the following two exact sequences: (1) 1 -> FxU(m)Foo+ -> F A X -> Cl(m) -> 1, (2) 1 -> U(w)nF A (w) -» FA(lit) -» /(w) -» 1. Proof. We first prove the exactness of the second sequence. The projection map is given by x h-> xO. If a- IIvPv 6 ^ is an element of I(m), then e(v) = 0 if pv divides m. Since Oy is a valuation ring, every ideal is principal, and hence we have p v e ^ = x v 0 v for some xv G F v x . We define xv as above if v is finite and pv is prime to nu We simply put xv = 1 if either v is infinite or & divides m. Then a=xO and thus the map F A O ) -> I(m) is surjective. If xO = O, then e(xv) = 0 for all v and hence x e U(O). Since xv = 1 if either v is infinite or py divides m by the definition of FA(W), we see that x G U'(w)flF\{m) if xO = O. This shows the exactness of the second sequence. Since uxO = xO for u e U(m)Foo+, we also know the exactness of (3)

1 -» U(m)Foo+ -> U(m)F A (m)F oo+ -> /(m) -> 1.

By definition, we have U(/rc)FA(m)Foo+ = {x G FA X | x p G Uv(»t) if p\ m and Xoo e Foo+}, and hence U(m)FA(/n)Foo+riFx = P(m). Thus we see

because iP(w) = P(m)/E(m) and E(w) is contained in U(m)Feo+. By definition, F x is dense in F m x = Tlp\ JF 7 ^. Thus, for each idele x e FA X , there exists a G F x such that (a"1x)mG HpOpx. That is, aAx0 is prime to m. Thus we can find y e U(WI)FA(WI)FOO + such that a'lx0 = yO. That is, oc^xy'1 G U(0)F oo+ . Since XJ(O)fU(m) = n ^ | m(Owx/\Jy(m)) = n ^ | m(Ojmv)x = (O/m)x, we can find ye O such that y = a ^ x y " 1 mod U(m), x G yayU(m). Therefore F A X = FxU(m)FA(/n)Foo+. Since

we see that

and hence

8.2. Hecke characters as continuous idele characters

247

FAx/FxU(i»)F«H. S = U(ifi)FA(w)F^{F x nF A (w)U(w)F o . + }U(w)F« + s Cl(ift). This shows the exactness of the first sequence. Let X* be a Hecke character modulo m; that is, X* is a character of I(m) such that A*((oc)) = oc^ if a e P(m), where § = X ^ a G Z[I] and a 5 = UGa°^ ae

for

x

F .

Theorem 1 (Weil). There exists a unique continuous character X : -> C x such that X(x) = X*(xO) if x G Xoo e Foo+, w/zere conjugation c.

x

a

= xv

and x

FA(*K)

oc

FAX/FXU(7TC)

0«d Mx~) = *~^ = n a x" a ^ a if

= x v for

o = Gv

and complex

Proof. By the exact sequence (3), we know that

Thus we can define a character A,: \J(m)FA(m) -> C x by X,(x) = A*(xO) for x e UO)F A (/tt). We extend this character X to U(w)F A (w)F oo+ by X(x) = A,*(xfO)x~^ where x"^ = ITax"a^a and Xf is the projection of x to U Here note that x f 0 = x0. Thus for a e P(i») = U(m)FA(m)Foo+nFx, A,(ax) = A,*(axfo)(ax)"^ = A.*((a))A*(xfo)(x'5x"* = ^*(xfo)x"^ = X(x). Now we extend this character to F A X = FxU(7rx)FA(/n)Foo+ by X(jx) = X(x) if y e F x and x E U(/n)FA(w)Foo+. This is well defined. Indeed, if yx = 8y for x,y G U(w)FA(m)Foo+ and y,8 G F X , then F B yl8 = y -1 x G U(m)FA(m)Foc+ and yS"1 G P(W). Then X(yx) = X(x) = ^ ( y ^ y ) = X(y) because X(ax) = X,(x) if a G P ( O T ) . This shows the existence of X. The uniqueness is obvious because

Exercise 1. Show the continuity of X as above. (A sequence xn e F A X converges to x if and only if for any ideal a of O and e > 0, there exists a positive number M such that if n > M and m > M , then x n -x m G U(a)Foo+ and I x n -x m I v < £ for all infinite v.)

248

8: Functional equations of Hecke L-functions

Exercise 2. Show that if t, = 0, then X* is a character of Cl(/n) and X is just the pullback of X* : CIO) —> C x by the isomorphism FAx/FxU(w)Foo+ = Cl(m) given in Lemma 1. Exercise 3. Let X be as in the theorem. (i) Show that ^(x) is contained in a finite extension K of F for all x E F A (m). (ii) Enlarging K if necessary, we suppose that K contains all conjugates of F over Q. Fix one finite place v of K and write p (resp. | | v) for its residual characteristic (resp. the v-adic absolute value on K). Then show that every place w of F over p is given in such a way that I x | w = | x a | v for some d e l . Write this place as w(a). (iii) If w = w(o), then a : F —> K extends to a : F w -> Kv by continuity. Define for x e F p x = ( F ® Q Q p ) x = n w | P F W X , x^ = n a x a ^ a where x a = (x w ( a )) a . Then show that there is a unique continuous character Xp : FAx/FxU(w)Foo+ -> Kv such that Xp(x) = X*(xO) if x e FA(/np) and ^p(xp) = X(xp)Xp"^ for all xp e F p x . Now for a given character X : FAx/FxU(/n)Foo+ -» C x such that ^(Xoo) = Xoo"^, we can recover a Hecke character X* by putting X*(xO) = X(x) for x e FA(m). In fact, if xO = yO for x and y in FA(m), then xy"1 e \J(m) and hence ^(xy' 1 ) = 1, that is X*(xO) = X*(yO). If a e P(m), then X*((a)) = X(am) for the projection am of a to FA(w). On the other hand, the projection a m of a to IIvlmFv is contained in \J(m) because U(/rc)FA(7rc)Foo+nFx = P(w). Thus ^ ( a j = 1. On the other hand, 9L((XOO) = = a"^. Since X is trivial on F x , we see 1 = X(a) = Xia^Xia^Xia^) X*((a))a'^ and hence X*((a)) = ofi if a e P(m). Thus the correspondence X <-> X* is bijective for a given infinity type ^ and for a given defining ideal m; that is, we have Corollary 1. The correspondence of Hecke characters: X C x with A,(Xoo) = x ^ for % e Z[I] instead of ideal characters. Exercise 4. Let m and n be two ideals. Show that XJ(m)\J(n) = U(m+n).

8.3. Self-duality of local fields

249

For a given Hecke character X as above, if X is trivial on U(w) and XJ(n), then by Exercise 4, X is trivial on U(wri-n). Thus there exists a largest ideal c such that X is trivial on U(c). This ideal c is called the conductor of X. We now study additive characters of FA- We start with the simplest case of A. Pick a prime p. Considering the p-fraction part [z]p of z e Z p , we define the standard additive character (4)

e p : Q p -> T = {z e C | | z | = 1} by e p (z) = exp(-27C V^lMp).

For any finite idele x e Af, we define ef(x) = n p e p (x p ) = exp(-27i V~-IZ p [x p ] p ). This is well defined additive character since for almost all p x p e Z p and hence ep(xp) = 1. We define eoo(Xoo) = exp(2rc V-T Xoo). Thus we have an additive character e : A —> C given by e(x) = ef(xf)e(Xoo). If a is a rational number, then a-Xp[a]pe

ZflQ = Z.

Therefore e(a) = exp(-27c V z lE P [a] p )exp(27cV :: Ta) = exp(2^z^[-[(a-Tp[a]v)) = 1. Namely e is trivial on Q and hence is a non-trivial additive character of A/Q. Theorem 2. For any number field F, there exists a non-trivial additive character e = e F : F A /F -> T . Proof. We have explicitly constructed e = eQ when F = Q. For a general number field F, we simply define ep by e?(x) = eQ(Trp/Q(x)). Since the trace map sends elements in F onto Q, ep is a non-trivial additive character of FA/F.

§8.3. Self-duality of local fields We fix a place v of F and consider the completion T = {zeC

K = F v . Let

| z | = l } , which is a multiplicative group. We want to show that

the group of continuous additive characters Homconti(K,T) of K is canonically isomorphic to K. We start proving this first assuming K = R. If a : R -» T is a continuous homomorphism, we consider its kernel L = Ker(cc), which is a closed subgroup of R. Suppose that it has an accumulation point x. Then x itself belongs to L, and we can pick y e L arbitrarily near to 0 but not equal to 0. In other words, z = y-x is arbitrarily near to 0 but not equal to 0. Then Zz is a subgroup of L whose adjacent elements have distance I z |. Since we can make | z | —> 0 keeping z in L, we know L is dense in R. Since L is closed, L = R and a is the zero map. Thus for non-trivial a, L is a discrete subgroup of R. Let z be the element in L-{0} with the minimum distance to 0. Then by the euclidean algorithm, for each x e L, we can find q e Z so that

250

8: Functional equations of Hecke L-functions

x e [qz,(q+l)z). Then x-qz e L has absolute value smaller than z, and hence by the minimality of the distance between z and 0, we know that x = qz. Namely L = Zz. Since the homomorphism ( p : x h ) exp(27iV-Iz"1x) has the same kernel as a, there is a unique automorphism P : T = T such that a = p°q>. First we fix the lifting i : T-{-l} -> ( - £ , £ ) by i(exp(2rc V ^ x ) ) = x for x with I x | < j . Since p is an automorphism, p sends a small neighborhood U of 0 onto another neighborhood U' of 0. We take a still smaller neighborhood W in U so that W+W c U. Thus pf(x) = i(p(exp(27i-vcTx))) is a well defined additive map on Wo = i(W), which is a small neighborhood of 0 in R. Since any additive map on WoDQ is induced by a linear map x h-> bx for some b e R , the continuity of p' tells us that pf coincides with x h-> bx for some b e R in a small neighborhood Wo. Then by the linearity of p, we see that p(exp(27cV~Ix)) = exp(27tV--lbx), which is an automorphism of T, and hence b = ± 1 . This shows that a(x) = exp(±2rc V - l z ^ x ) . We have shown the surjectivity of the natural map R B b H eb G Hom con ti(R,T) given by eb(x) = exp(2rc V-l bx). The injectivity of this map is obvious. Thus we have Hom cont (R,T) = R via eb <-> b. By this, HomCOnt(R2,T) = R 2 and hence Homcont(C,T) = C. We can check that this isomorphism is induced by e b <-> b for e b given by e b (x) = exp(2n;V~lTrc/R(bx)). We now extend this result to any finite place v. For that, we recall the duality theory of locally compact groups. Let G be a locally compact abelian group. We consider its dual group G* = Homcont(G,T) and on it we put the topology of uniform convergence on every compact subset of G. Thus a sequence of characters a n : G -> T is convergent to a character a if a n converges to a uniformly on any compact subset X of G. Let e : K -» T be the additive character defined in the previous section, i.e. e(x) = exp(-27C-v/-T[TrK/Q (x)]) if the residual characteristic of v is p and e(x) = exp(27cV-T[TrK/R(x)]) if v is infinite, where [x] is the p-fractional part of x. Then we can define a pairing ( , ) : K x K -> T by (x,y) = e(xy). Then we want to prove Proposition 1. The above pairing induces HomcOnt(K,T) = K. Before proving the proposition, we prove a preliminary lemma:

8.3. Self-duality of local fields

251

Lemma 1. Identify T with R/Z and let Ik be the image of the interval (-—,—) in T for positive integers k. If a homomorphism a of a group G into T satisfies l\ 3 oc(G), then a = 0. There are two consequences of this lemma: (i) If G has a system of neighborhoods of the identity consisting of (open) subgroups H, then any continuous homomorphism a : G —» T becomes trivial on a sufficiently small open subgroup H. (ii) If G is compact, then G* is discrete. In fact, if a continuous character ocn of G converges to a in G*, then the convergence is uniform on G (because G itself is compact) and oc-an has values in Ii on G for sufficiently large n. This means (Xn = a and thus G* is discrete. We add one more remark: (iii) If G is discrete, then G* is compact, because G* is easily seen to be the closed subspace of compact space T G , which is the set of all functions of G having values in T. Proof (Pontryagin). We claim that if jt e l\ for all j = 1,2,3,-.., k, then t e Ik. We prove this by induction on k. If k = 1, the assertion is trivially true. Suppose that the assertion is true for k-1 and try to prove the case of k. Choose a representative x of t in (-«,, (-—,—), there is nothing to prove.

n,

-ZTT——). If x is already in

Thus we may suppose that

| x | e [g£, 30^1))- Then | k x | e [ - , - ^ - ^ - ) . This interval is inside [0,1) and hence kt e Ii means that kx e (-^-, ^-) and hence x e (-—,—), which 3

J

3K JK

means t e Ik. Now let us prove the lemma. If oc(G) is contained in Ii, then a (kg) = k a ( g ) E Ii for any positive integer k and any g e G. Thus Ik D a(G) for all k, which means that oc(G) = 0 because D i A ^ l O } Proof of Proposition 1. We may suppose that v is a finite place over a rational prime p. First assume that K = Q p . If <() e Hom cont (Q p ,T), then for a sufficiently small neighborhood V of 0 in Q p , Ii z> (|)(V) by continuity of <|). We can take the subgroup p r Z p as V. Thus Ii z> (|)(prZp) and hence by the lemma, 0(p r Z p ) = 0. Since for any x e Q p , we can find a sufficiently large exponent k such that pkx e p r Z p , we see pk(|)(x) = <|)(pkx) = 0. Thus identifying T with R/Z via xh-»exp(27tV-Tx), the value of continuous character is contained in the image of fractions whose denominator is a p-power, i.e. T p = Q p /Z p z> <|)(QP)

252

8: Functional equations of Hecke L-functions

(see (1.1)). Thus what we need to show is Homzp(Qp,Tp) = Q p under the pairing (x,y) = xy mod Z p , because any continuous homomorphism <|): Q p —»Tp is Z p -linear. Thus Q p is sent into H o m z p ( Q p , T p ) via X H (()X for $x(y) = (xy mod Z p ). Since T p is divisible (i.e. for any y G T p , we can find x G T p such that px = y), applying the functor Homz p (*,T p ), we have the following commutative diagram: QP

4a

ip

0 -> Hom Z p (T p ,T p ) -> Hom Zp (Q P ,Tp) where both rows are exact ((1.1.1a)), a takes ze Z p to multiplication by z on T p = Qp/Zp, p(x) = (|)x and y takes t G T p to the unique homomorphism <[) such that <|)(1) = t. The surjectivity of 8 can be proven as follows. If <>| : Z p -» T p is a Zp-linear map, then we can extend it to p" r Z p by putting (|)(p"r) = xr for xr in T p such that prxr = $(1). Thus <>| is extensible to p"rZp for any r and hence extensible to Q p . By definition, y is an isomorphism. Thus we only need to show that a is an isomorphism in order to show that p is an isomorphism. If oc(x) = 0, then multiplication of x on p"rZp/Zp = Z p /p r Z p is zero and hence x is divisible by p r for arbitrary r. This shows that x = 0 and hence a is injective. If ty : T p -> T p is a Zp-linear map, then $ induces a map (t>r: p' r Z p /Z p -> p" r Z p /Z p . Thus <|>n G End(Z/p r Z) = Z/p r Z. Since <|>r induces (t>s if r > s , as an element of Z/p r Z, <|>r = <|>s mod p s . Picking an integer x n such that xn = <|)nmodpn for each n, the sequence {xn} converges to x G Z p because | x r -x s | p < p"s if r > s. By definition oc(x) = 0. Thus a is surjective and hence the assertion follows when K = Q p . By this we know HomZp(Q p r ,T p ) = Q p r via the pairing (x,y) = ZJ =1 XJVJ mod Z p . Now we treat the general case. Identify K with Q p r by choosing a basis {vi,...,vr} over Q p . We can also identify the Qp-dual vector space K* of K with K via the pairing (x,y) = TrK/Qp(xy). Thus by choosing the dual basis Vj* of K (i.e. (vk,Vj*) = 8kj), we can identify K with Q p r . Then by the above argument, we know that Hom cont (K,T p ) = K via the pairing (x,y) = exp(27cV-lTrK/Qp(xy)). This shows the proposition. Theorem 1. Define a pairing ( , ) : FA X F A -» T by (x,y) = e(xy) for the adelic standard additive character e : FA/F —> T. Then this pairing induces an isomorphism FA = Hom cont (FA,T).

8.4. Haar measures and the Poisson summation formula

253

Proof. Since FA = FAfXFoo and Foo is a product of finitely many copies of R and C, we only need to prove the self-duality for F Af . Since n(5 for positive integers n gives a system of neighborhoods of 0 in FAf, for any continuous homomorphism <>| : FAf -» T = R/Z, we can find n so that (|)(nd) = 0 by using Lemma 1. Thus we know that Q/Z z> (|>(FAf). We first assume that F = Q. As seen in (1.1) and (1.3b), e f : A f /Z f = 0 p (Q p /Z p ) = Q/Z. Using the commutative diagram analogous to the one in the proof of Proposition 1, 0

>

Z

> 0pTp

> Af

la

i (3

> 0

4 7

0 -> Hom(Q/Z, Q/Z) -> Hom(A f , Q/Z) -> Hom(Z, Q/Z) -> 0, 8

we know that Hom(Af, Q/Z) = Af. This shows the assertion for Q. Now we can take a basis vj of O over Z; then F = Q d and O=Zd (d = [F:Q]) via this basis. Then we see that FAf = Afd. The character e takes, by definition, FAf into Q/Z (in fact e = eQ©Tr for the trace map Tr : F Af —> Af). Thus we know from the same argument as in the proof of Proposition 1 that Hom(FAf,Q/Z) = FAf via the pairing induced from e. This finishes the proof. By Lemma 1 and the above proof of the theorem, for each non-trivial continuous character <>| : F v —> T (resp. <|) : FA -> T), there is a fractional ideal pv~5 in F v (resp. -ft"1 in F) maximal among the ideals a with (|>(a) = l. The inverse of this ideal is called the different of <(>. If e is the standard character, then the inverse of the different of e is the dual lattice of O under the pairing (x,y) = Trp/Q(xy) and hence is the different inverse &l in the classical sense. If <|)(x) = e(ax) for all x e FA, then the different for (J) is given by aft. Since the pairing ( , ) on FA is continuous under the topology on FA, the isomorphism of Theorem 1 is in fact an isomorphism of topological groups. Exercise 1. Show that the isomorphism of Proposition 1 is continuous if

§8.4.

Haar measures and the Poisson summation formula

To prove the functional equation for Hecke L-f unctions of number fields, we need the theory of Fourier transform on a locally compact abelian group G, which we explain now. We assume the following conditions. (la) There is a continuous pairing

(,):GxG—>T = { z e C |

|z|=l}

254

8: Functional equations of Hecke L-functions

which induces an isomorphism G = Homcont(G,T) (on the right-hand side, we put the topology of uniform convergence on all compact subsets of G and this isomorphism has to be an isomorphism of topological groups). (lb) G has a cocompact discrete subgroup F. Here the word "cocompact" means that G/T is compact. (lc) The orthogonal complement F* = {x* G G\ (X*,X) = 1 for all x G F} is again a cocompact discrete subgroup of G. In fact, the third condition is superfluous and it is known that it follows from (la,b) (see the remark after Lemma 3.1). In our application, G will be a vector space over R or the adele ring F A . Let us check the conditions (la,b,c) for a real vector space V = R r . Let S : VxV -* R be any non-degenerate inner product and L be a lattice of V. Then L is a discrete subgroup of V and by choosing a basis {vi,,..,v r } of L over Z, we can identify L with Z r . Since the Vj's form a basis of V over R, we can identify V with Rr via this basis. Then V/L = (R/Z) r is compact. Now we define ( , ) : V x V - 4 S by (x,y) = exp(2jcV = lS(x,y)). Then L* = {x* G V | (x*,x) = 1 for all x G L} = {x* G V | S ( X * , X ) G Z

for all

XG

L}.

Then L* is the dual lattice generated by the dual basis Vi* such that S(vi*,Vj) = 8y. Thus L* is again a lattice and is discrete and cocompact. When G = FA, we can take F as a discrete subgroup of FAOn the group G, there exists a Haar measure dji with values in R, which satisfies the following conditions: (2a) JLL is defined on a complete additive class containing all compact subsets of G (e.g. a union of countably many compact subsets is measurable); (2b) 0 < |Li(K) < +oo for all compact subsets K o / G (|n(K) > 0 if the interior of K is not empty), ji(U) = SUPUDK compactM-(U) for all open sets U and \x(X) = Infuz,x, u openM-(U) for all measurable subsets X; (2c) ja.(x+X) = [i(X) for all measurable subsets X and x G G. Under the conditions (2a,b,c), the Haar measure is unique up to a constant multiple. The uniqueness is intuitively obvious because if K is a disjoint union of K' and x+K', then |i(K') = |i(K)/2. In this way, if one fixes a compact neighborhood of 1 with positive measure, by subdividing it, the measures of all compact subsets are uniquely determined. The condition (2c) implies the equality

8.4. Haar measures and the Poisson summation formula

255

Jef (x+y)dji(x) = fGf(x)d*i(x)

(3)

for any integrable function f with respect to JLL. When G is a real vector space, the Haar measure on V is a constant multiple of the Lebesgue measure induced by the identification G = R r . Now we fix the Haar measure (I on G and consider the Hilbert space L2(G/T) of /^-functions on G/F. In fact, by taking the fundamental domain K of F in G so that K is compact and the complement of K is open, the measure |i induces a measure on K, which gives a Haar measure on G/T. By multiplying by a suitable constant, we may assume that Jo/rdM-W = *• Th e Li space Z^G/F) is the space of functions x

2

X < +o

JG/T ' ^ ) ' ^M-( )

f: G/F -> C

°- Thus

we can

which are square integrable; i.e,

define the positive definite hermitian inner

product on Li(G[T) by

For y* e F*, we consider the character \\f = \|/y* : G —» T given by \j/T*(x) = (y*,x). Then for y e T, \|f(x+y) = (y*,x)(y*,y) = \\r(x) because (Y*,Y) = 1. Thus \|/ is a continuous character of G/F. Since G/F is compact and \|/ is continuous, we know that \}/ e L2(G/F). Moreover =

L

Since G = Hom c o n t (G, T), if y*,8* e F * and y* * 8*, then for some y e G, Vy*(y) * V8*(y)« On the other hand, we consider an operator T y : L 2 (G/F) -> L 2 (G/F) given by Tyf(x) = f(y+x). Then we see from (3) that = J G / r f ( y + ^ ) g ( x ) ^ ( x ) = J G / r fWg(x-y)d|l(x) = . This implies 1

=
and hence (xj/^,^*) = 0 if y* * 8*. Moreover it is known (see [P]) that (4) \|/yn for y* e F* gives an orthonormal basis of L2(G/F). Thus, any f e L2(G/F) is expanded into the /^-convergent series f

= Xy*er* C(Y*)VT*

for C

(Y*)

256

8: Functional equations of Hecke L-functions

Exercise 1. By applying the functor Hom(*, T) to the exact sequence of abelian groups 0-»M—» N -» L -> 0, we have naturally another sequence (5)

0 -> L -> N -> M -> 0,

where M denotes Hom(M, T) for any abelian group M. Show that the sequence (5) is exact. For the surjectivity of the last arrow: For each a e M, consider the set A of pairs (X,£) consisting of subgroup X of N and a homomorphism E, : X -> T such that ^ | r = ex. Put an order (X,£) > (X',£') on A when X D X 1 and £ I x1 = £'• Then by Zorn's lemma, there exists a maximal element (X,^) in A. Show that X = N and the image of Z, in M is a. Now we define the Fourier transform for any integrable function f on G by

(6)

J(f)(y) = Jr(0(y) = JGf(x)(x,y)dn(x).

Since | (x,y) 1 = 1 , the above integral is always convergent if f is integrable. Theorem 1 (the Poisson summation formula). Suppose that (i) f is a continuous function on G integrable with respect to \i, (ii) Z YGr f(x+y) and Zy*Gr* jF(f)(x+y*) are both absolutely and locally uniformly convergent. Then we have

£ y s r f(x+y) = Y Inparticular,

^yeT

f(y) =

Proof. The function O(x) = Z 7 e r f(x+y) is invariant under translation by the elements of T. Thus we may consider <& as a function on G/T. Since G/T is compact, O is square integrable. Thus we can expand C(Y*)\J/Y*(X) for

c(y*)

=

Let K be the fundamental domain of G/T we have already chosen. Then

,> = J G/r (-y*,x)O(x)dn(x) = J G / r (-Y*,x)I y e r = E v e r JG/r(-Y*'x)f(x+Y)d|4(x) = JUYK+7(-Y*,x)f(x)dn(x) = JG(-Y*,x)f(x)d^(x) = Thus we have

8.5. Adelic Haar measures

257

Since the convergence is locally uniform, the above identity is not only in the L2-space but the actual identity of the two functions valid everywhere. §8.5. Adelic Haar measures Now we want to construct explicitly the Haar measure on FA and FA X . When v is infinite, F v is C or R and thus we have the Lebesgue measure dx on F v . The Haar measure on the multiplicative group F v x in this case is given by I x I-1dx when v is real and djivx(z) = |x 2 +y |"*dxdy (writing z = x+V~Ty) when v is complex. Now we treat the case where v is finite. We only need to make explicit the integration of locally constant functions on Fv for finite v under the Haar measure. A function f on a topological group G is called locally constant if for any x e G , there exists an open neighborhood V of x such that f is constant on V. Thus for a given x e V, the set ( y e G | f(y) = f(x)} is an open set. In particular, any locally constant function is continuous. To know the volume of open compact subsets of the Haar measure on CV, we may assume that JI(OV) = 1. Since {jp^}j=i,2,... gives a system of neighborhoods of 0 and Oy = Uaa+pv-" is a disjoint union of open subsets, where a runs over a representative set for CK/pyK Thus we must have

This shows that for any generator G5V of (1)

j

Of course this formula is valid for all a e F v and i e Z (not necessarily positive). If f: F v —» C is a locally constant function, then as already remarked, the set f^fCx)) = {y e F v | f(y) = f(x)} is an open set for a given x. Thus we can write, for f(x) = c, f -1 (c) = UaCa+jv1^) as a disjoint union. Then formally and ji v (f x (c)) = 2 >

^

If the above sum is absolutely convergent, then f is called integrable. In particular, if f is compactly supported, that is, the closure of the set ( x e F v I f(x)*0} (which is called the support of f) is compact, then f is integrable. We then have the property (4.2a-c) for this Haar measure. Similarly we can define the multiplicative Haar measure d|Lix by

25 8 (2)

8: Functional equations of Hecke L-functions \i

- I ) ' 1 if j > 0

for a e O v x . Since a+/v* = a(l+;v*) and the subgroups {l+pvJ}j>o give a system of neighborhoods, the same argument as in the additive case shows the invariance property of |ivXExercise 1. (i) Show that ( x e F v | f ( x ) * 0 } is closed if f is compactly supported and v is finite. (ii) Show that for any locally constant function f whose support is in F v x , f

*

if

I

I 1

\v xf(x)djiv (x) = N(jtfy)(N(pv)-l) Jp f(x) I x | v d|i v (x). J Jrv

•

rv

(Reduce the problem to the formula Jiv^a+pv1) = (l-Nipy)'1)'11 a I ~ M-vfa+^v1)-) (iii) Show that for any compactly supported locally constant function f on F v and a * 0, J Fv f(ax)d|iv(x) = I a | v"1JFvf(x)djiv(x) (use (ii)). We now define the Haar measure on FA. For each fractional ideal a, we write & for the closure of a in FAf, which coincides with Ylv fmite^v- First, on FA^ we define the additive measure |if and the multiplicative measure |Hfx by |U.(a+a) = N(a)~l for each fractional ideal a in F, ^i (aU(a)) = (U(0):U(a)) 4 = WQ/a)*)'1 for each ideal a in a x

Since {U(a)} a (resp. {&}a) gives a system of neighborhoods on AfX (resp. Af), we can verify that |Lif and |ifX are well defined. Let [U (resp. |ieoX) be the additive (resp. multiplicative) Haar measure on F«> induced by the Lebesgue measure as already defined. Then if a function is of the form 0(x) = <|)f(xf)oo(Xoo) with a locally constant function % on FAf and an integrable function L, on F«», then the integration under the Haar measure \x = |if<8>}ioo (resp. can be computed by

JLLX

= |iXf0jiXoo)

JFAf<

= JFAf<))f(xf)d|if(xf)xJFeo(|)oo(xee)djieo(x<>o). Exercise 2. Show that (if = ® v fmiteM-v. First show that the problem is reduced to proving \if{a) = Tlv finiteM'v(flv) for all integral ideals a and then show that the infinite product of the right-hand side is in fact a finite product and gives the volume of the left-hand side.

8.5. Adelic Haar measures

259

Examples of integrals: Orthogonality relations (Lemma 2.3.1): Let % a n d A, be continuous characters (with values in T) on G = Ovx or Ov. Let JJ, be a Haar measure on G. Then

Proof. Since %^ is continuous, it becomes trivial on a sufficiently small subgroup H of finite index by Lemma 3.1. Thus we see (3) By Lemma 2.3.1, we see that 2,XEG/HXMX)-|(G:H)

.f

x

=

r

i

>

Then the assertion follows from the fact that |i(G) = |i(H)(G:H). Gauss sum. Let % be a continuous character of F v x and $ be a non-trivial additive character of F v . Let G5~r0v = fty1 be the different inverse of <>| for a generator 03 of pv (i.e. G5"rOv is the maximal fractional ideal in F v such that (j)(G3"r0v) = 1) and G3f0v be the conductor of % (i.e. G5fOv is the maximal ideal such that %(l+GJfOv) = l if % is non-trivial on 0 / and m = 0 if % is trivial on OvX). When f > 0, we consider the integral

I G3r |

(4a)

JG3Ov

This is a Gauss sum. In fact, by the variable change: x i-» 03

x, we see

I C5r | v x(B3 r+f )J 05 . r . fOvX x
mO d

f rf f rofX(x)JOv(|>(C5- - (x+03 y))dn(03 y)

= ^ x mod

^

which is visibly a Gauss sum. We only integrate %<)| on G5"f"rOvX but the same integral on G5mOvX vanishes if m ^ -f-r. In fact, we have (4b) Proof. We see that

J03mOvX%(|)(x)d|i(x) = 0 if m*-r-f and f > 0 .

260

8: Functional equations of Hecke L-functions

First suppose that m < -r-f. Then the above integral is equal to

| C3 | v £

x mod

S 5 f X

(

v

We then compute

J C 3 f o v * ( G 3 m ( x + y ) ) d ^ ( y ) = I G5f I v<]>(G5mx)JOv(G3m+fy)d|J.(y). Since m+f < -r, y h-> <|)(G5m+fy) is a non-trivial additive character. Hence by (3), we see the right-hand side of the above integral vanishes. Thus we have the vanishing of (4b). Now we suppose that m > -r-f. Then

= X x mod

raf-iJx+05f-iOvX(x(l+03

f 1

- y)Wn3mx(l+05f-1y))d^(x(l+05f-1y))

= Zxmod®M I xG5M I v$(03mx)%(x)Jn5f.1OvX(l+ti3f-1y)dH(y). Since Jn5f.ic,vX(l+O3f'1y)d|i(y) = l^(G5 f a)I yel+ro f-i Ov/1+03 f-i Ov X(y) = n(G3fOv)J1+05f.iOvX(y)d^x(y) = 0 by (3), the vanishing of the integral we wanted follows. Integrals at °°. Write I for the set of all embeddings of F into C. We consider the inner product S : FxF -> Q given by Trp/Q(xy). This inner product extends to V = Foo as a real bilinear form. We identify Foo with R a ( R ) xC a ( C ) . Then for each a e F, the image of a in V is given by ( a a ) a € a(C)Ua(R> We consider the function \|/ y (v) = exp(-7tZ a € l y a I v a I 2 ) for y o € R+ with y a = y a c for complex conjugation c, where for a e a(C), we agree to write v a c for v a . Then we see that for any polynomial P(v), \|/p(v) = P(v)\j/(v) is integrableon V and we have Lemma 1. Let du be the usual Lebesgue measure on V. Then (5a)

J FeoVy (u)e(Tr F/Q (uv))du = 2"W(y)-1/2\i/y-i(v),

where e(x) = exp(2TcV-lx), t = #(Z(C)) and N(y) = UGeiyo. Moreover let 0 < r| e Z[I] such that T] a =0 or 1 for a e a(R) and r\oV[cc=0 for a E a(C). Then we have

8.6. Functional equations of Hecke L-functions

261

J Foo uV y (u)e(Tr F/Q (uv))du = 2-W(y)- 1/2 y^i^ } v^>

(5b) where

[r]} = Zael'Ha e Z.

Proof. Note that Trp/Q(uv) = Zaea(R)UaVa+Z
exp(-7cyu )e(uv)du and J— oo

exp(-27iyuu)e(uv+u v)du. J—ooJ—oo

The first integral is equal to -y/y exp(-7ix2/y), and the second follows from the first immediately. We obtain the last assertion via the differentiation by j - \ = n ae 11 g^" I

of the first formula (5a).

Exercise 3. Give a detailed explanation of the computation of the integral (5b,c).

§8.6. Functional equations of Hecke L-functions In this section, we prove the functional equation for Hecke L-functions via the method of Tate-Iwasawa. We basically follow the treatment in [W1,VIL5]. We first deal with the Fourier transform of standard functions on F v or FA. For a finite place v, a function f on F v is called standard if it is locally constant and compactly supported. For a infinite place v, a function f is called a standard function if f(x) = cxAexp(-7Cx2) for a constant c and A = 0 or 1 when F v = R, f (x) = cx A xBexp(-27i I x I ) for a constant c and integers A > 0 and B > 0 with AB = 0 when F v = C. A function f on FA is called standard if f(x) = Ilvfv(xv) f° r a standard function fv on F v for all v and fv is a characteristic function of (\ for almost all v. Let G be either FA or F v . Let e : G —> T be the standard additive character. We define a pairing ( , ) : GxG -» T by (x,y) = e(xy). Then we already know that under this pairing, we have G = Homcont(G,T). Let \i be the additive Haar measure defined in the previous section. We modify JI as follows. Define

\i\

by

2JJ,V

if

ve

a(C),

jn'v = |Liv

for

ve

a(R)

and

|TV = I G3r | V 1 / 2 J I V for finite places v, where we write the local different T% = G3r0v f° r a prime element C3 of Ov. Then we define p,1 = ®|Ufv as the Haar measure on FA. We will see later IFA/F^H' = 1. We now define the Fourier transform of standard functions by

262

8: Functional equations of Hecke L-functions

When G = F v for a finite place v, then f is compactly supported and hence integrable. Thus f(f) is a well defined function on F v because | (x,y) 1 = 1 . When v is infinite, any standard function is integrable and thus the Fourier transform is again well defined. When G = FA, then by definition of standard functions, f(x) = foo(xoo)ff(xf) and ff is compactly supported and f<x> is integrable. Then we see

JGf(x)(x,y)d^i'(x) = and each integral of the right-hand side is well defined and hence the Fourier transform is a well defined function on FA- We also know that if G = FA and f = Ilvfv is a standard function, then

Thus the computation of Fourier transform of standard functions is reduced to local computation on F v for each place v. As seen in §5, we already know that (1)

tffv)

= J R x A exp(-7ix 2 )(x,y)d^'(x) = V^VexpC-Tiy 2 ) if F v = R,

T(U) = J c x A xBexp(-27C | x | 2 )(x,y)d^(x) = V = I A+B y B y A exp(-27r | y 12) if F v = C. Thus we compute the Fourier transform on F v for finite places v. Let K be a generator of the maximal ideal of C\ and G3rOv = $ v be the different of e. First we take the characteristic function O = O v of Oy. Then we claim (2)

n&)(y)=

lG3r|v1/2O(G5ry).

Let us prove this. Note that the additive character y h-> e(xy) is trivial on a, if and only if x e G3~r0v. Thus the orthogonality relation tells us that J F O(x)(x,y)dji(x)=J 0 e(xy)d|i(x) =« J v JUv ^ [0 if x ^ 03 O v , which shows (2), because |i' v = I G5r I v1/2M-v Since r = 0 for almost all v, ^•(Ov) = O v for almost all v. If f is a standard function on FA, then f = n v f v and fv = O v for almost all v, and we know from (1) and (2) that > = FIv7(U) is again a standard function.

8.6. Functional equations of Hecke L-functions

263

Now we take a continuous multiplicative character X of F v x and assume that X is non-trivial on OvX. Thus if G3f Q, is the conductor of X, then f > 0 (G3f CV is by definition the maximal ideal such that ^(l+G3fOv) = 1). Let O^ be the locally constant function defined as follows:

Then defining K = KV = | G5f | V"1/2JO x^"1(x)e(G5"r"fx)d|i(x), we have * * 0 ( y ) = 1®f+r I v1/2X(©^f)Kvx.i(©r+1y).

(3)

In fact, by the computation of the Gauss sum, we already know that

J Fv O x (x)(x,y)d|i(x) = 0 if yO v x * G3"r"fOvx, |05f+r|v1/2?l(03r+fy)Kv if yO v x = G5-r"fOvX, which shows (3). By (3), we know that ^(G5r+f)Kv is determined independently of the choice of G3 since the Fourier transform has nothing to do with the choice of G5. This number is called the local factor (or local root number) of X and it is not so difficult to show that | KV I = 1 by using the Fourier inversion formula (see Exercise 2.3.5, Corollary 1, and the proof of Theorem 1 below). Let O be a standard function and O' be its Fourier transform. Consider another function Oz(x) = O(zx) for z e FA X , which is again a standard function. Then by definition,

= JFAO(zx)e(xy)d^'(x) =

= |z|

This formula will be useful. Let X : FAx/FxU(c) -> C x be a Hecke character of conductor c and of infinity type £, = Za£a<3. We may assume that X has values in T by dividing A, by a suitable power of the norm character cos(x) = |x|^ (in fact, if Xoo = l,then |x|^ = N(xO)'s and L(s,A,cot) = Exercise 1. Show the existence of cos with the above property, i.e. show that there exists

SGC

such that I ^(x) | =

IXIA*

264

8: Functional equations of Hecke L-functions

Let Xv be the restriction of X to F v x . If F v = R and a : F -> R is the corresponding field embedding, then Xw(x) = x^ a | x | '^° = x"Av | x | A v for A v = 0 or 1 according to the parity of £ a . If F v = C , then Xv(x) = x- A v x" B v |x| A v + B v for integers Av > 0, B v > 0 and A v B v = 0 , because X has values in T. Exercise 2. Show that if % : C x —> C x is a continuous character, then there exist integers A > 0, B > 0 and AB = 0 and a complex number s e C such that %(x) = x"Ax"B | x | 2 s . Take an idele b such that bO- cb for the absolute different d of F and boo = 1. Then for each v, we may assume that b v = 03 in the above computation. We then define a standard function attached to X by Ox = IW>x,v as follows. If Xv is trivial on 0VX (this is the case for almost all v), we put ®XV = 3>v (the characteristic function of Ov) and if Xv is non-trivial on OyX, O^v is as in (3) and if v is real, O?tv(x) = xAvexp(-7ix2), and if v is complex, then = xAvxBvexp(-27i | x | 2 ) . Thus by (1), (2) and (3), we have (4) where K = n v K v and KV is as in (3) if Xv is non-trivial on Ovx, Kv = 1 if Xv is trivial on Ovx and KV = V-T Av for v real and KV = V-T Av+Bv for v complex. Now we define for any standard function O the zeta integral: (5)

Z(s,A.;*)

By definition, Z(s,A,;O) = IIVJF x^>v(xv)^v(xv)|xv|^d(iXv(xv). Now we compute this integral locally and show that it is essentially the Hecke L-function L(s,A,). We start computing the integral for finite places v. First suppose that O v is the characteristic function of CV and Xv is trivial on OyX. Then

(6) J

F v

| | ^

^=0 J t s j O v | v

J x d ^ x v = (l-X*(pw)N(Pvy*y\

8.6. Functional equations of Hecke L-functions

265

which is the Euler factor of L(s,X) at p,. For other finite places v, Supp(Ov) is contained in G3"nOv for a sufficiently large n. Thus taking the characteristic function % of this set, we have, for a sufficiently large constant M, | v s | < MJC(X) I x | v a for a = Re(s). Then (7a)

| J FvX O v (x v )?i v (x v )|x v | s v d^ x v (x)

If O v = O^v in (3), we can compute J F xOv(xv)Xv(xv) I xv I vsd|iv(x) explicitly. JF x In fact, we see that (7b)

J FvX Ox v (x v )X v (x v )|x v |^d^ x v (x) = J OyX d|i x v (x) = 1

because O^v = ^v"1 on CVX and 0 outside. Anyway we have (8a)

|JFAfXO(x)?i(x)|x|sAd^fX(x)| <M^F(c)

for a = Re(s),

which is convergent if a > 1. We conclude that (8b) Now we compute the integral at infinite places. Since the standard function Ov(x) is equal to either xAexp(-?cx2) or xAxBexp(-27C I x | 2 ) according as F v = R or C, <X>V decreases exponentially if | x: | —> +<*>. Thus if Re(s) is sufficiently large, the integral J

)

|xv|'djixv(x)

converges absolutely if Re(s) is sufficiently large. Thus Z(s,^;O) is a well defined analytic function of s if Re(s) is sufficiently large. Now we compute x^)v(xv)A,v(xv) I xv I vsdjj,Xv(x) for O v = O^v. First assume that F v = R and x) = x"A | x | A . Then O v = xAexp(-7Cx2) and we have (9a)

JFvXOv(xv)?iv(xv)|xv|svd^ixv(x) = J ^ exp(-7cx2) | x | ^ ^ J exp(-7cy)y(s+A/2)-1dy = 7i' (s+A)/2 r((s+A)/2) =

When F v = C and Xw(x) = xATB

| x |A+B,

266 (9b)

8: Functional equations of Hecke L-functions JFvx*v(xv)^v(xv) |xv|svdnxv(x)

= J FvX exp(-2* I x | 2 ) | x2 + y 2 1 - ( ^ ^ = 2- 1 (27i) 1 - (s+(A+B)/2) r(s+(A+B)/2) = We put Gj^(s) = IIv infiniteGxv(s). Then we have (9c) Thus finally we see that (10)

Z(sXOx)

= 2-lGxJs)L(s,X)

if Re(s) > 1.

Until now we have only used the fact that X is a continuous character of X F A X / U ( C ) . Hereafter we suppose that X(F ) = 1 and prove the functional equation. Let O be a standard function on F A and O1 be its Fourier transform. Since FA/F is known to be compact and F is a discrete subgroup of FA ( 1 3 C ) , we can apply the Poisson summation formula in §4 to this situation. We first review the formula. Let F 1 = {x e F A I (x,F) = 1}. Then F 1 = Homcont(FA/F,T). Since FA/F is compact, F 1 is a discrete subgroup of FA by identifying FA with Homcont(FA,T) (Lemma 3.1 and Theorem 3.1). Since the standard character e is trivial on F, (!]£) = e(r|£) = 1 for § and r\ in F. Thus F is contained in F 1 . Thus F^/F is a discrete subgroup of FA/F. Since FA/F is compact, F^/F must be discrete and compact. Thus F^/F is a finite group. Let x E F X - F and suppose N X G F (such an integer N > 0 always exists because F^/F is finite). Since F is a field of characteristic 0, we can find y e F such that Ny = Nx. Thus x-y is killed by N in FA- Since FA is torsion-free, x = y, a contradiction. Thus F 1 = F. Let G be a locally compact abelian group. We suppose that there is a pairing ( , ) : GxG —» T under which G = Hom cont (G,T). Let F be a discrete cocompact subgroup and F* = {x e G I (x,F) = 1}. Then F* is again a discrete cocompact subgroup. Let |J, be a Haar measure on G. Let K be a fundamental domain of G/T and normalize [i so that jKd|J, = 1. Then the Poisson summation formula reads

if both sides are absolutely convergent and the Fourier transform ^F(f) with respect to JI and ( , ) is well defined (i.e. f is continuous and integrable on G). Let us apply this formula for f = O and G = FA, F = F* = F. Thus we need to

8.6. Functional equations of Hecke L-functions

267

show that Jp^dM-' = 1. By (1.3c), we have a canonical isomorphism Thus JFA/F^M- ~ JFoo/o^M-00* -^et {a)i»-*»»c°d} (d=[F:Q]) be a basis of O over Z. We identify Foo = C a ( C ) xR a ( R ) and C with R 2 taking the basis (1,V-1). Thus COJ can be considered as a vector in F^ = ( R 2 ) a ( C ) x R a ( R ) whose component at each a e a is given as follows: For c e a ( C ) , (Re(c0ja),Im(c0ja)) and for a e a(R), cof. Then ~ = I det(coi,...,cod) | = 2"11D | 1 / This shows that |i' on FA is the right choice, i.e., Theorem 1. Let <$> be a standard function on FA and ' te /& Fourier transform. Then the function s h-> Z(s,X,;O) defined by (5) when the integral is convergent can be continued analytically as a meromorphic function on the whole complex plane. It satisfies the following functional equation:

If one specializes O in the theorem to Ox, we derive from (4) and (10) the following result: Corollary 1. L(s,X) can be continued to a meromorphic function on the whole s-plane and satisfies the following functional equation:

)( ID | W( where c is the conductor of X, D is the absolute discriminant of F, b is the idelefixedin(3)and K is the root number for X. Before proving the theorem, we start with the following lemma in [Wl, VII.5]. We can decompose FAX/FX = ( F £ } / F X ) X R + via X H (xf(Xco|x^1/[F:Q]), |x| A ). We know that (F^/F x ) is a compact group (Theorem 1.1). Lemma 1 ([Wl, VII.5]). Let Fi be a measurable function on R+ with 0 < Fi < 1. Moreover suppose that there exists an interval [to,ti] in R+ such that Fi(x) = l if x < to and F i ( x ) = 0 if x > t i . Then the integral f(s) = J Fi(x)xs"1dx is absolutely convergent for Re(s) > 0. The function f(s) can be continued analytically in the whole s-plane as a meromorphic function. Moreover f(s)-s"1 is an entire function. If Fi(x)+Fi(x"1) = 1 for all x e R+, then f(s)+f(-s) = 0.

268

8: Functional equations of Hecke L-functions

Proof (A. Weil). First take as Fi the function $ such that <|)(x) = 0 if x > 1 and

(x) = 1

if

x < 1.

Then

f(s) = j l xsAdx = [xs/s]J = s"1

if

Re(s) > 0 and the assertion is obvious. For general Fi, we see that

fCs)^"1 = J~ (F^Xx^dx. Since F^-<() is a bounded measurable function with compact support on R+, the above integral is (locally) uniformly and absolutely convergent for all s, which gives an entire function of s. Note that (|)(x)+(|)(x"1) = 1. If F1(x)+F1(x"1) = 1, then F2 = Fi-<() satisfies F2(x-1) = -F2(x). Thus replacing x by x"1 in the above integral, we see f(-s)-(-s)"1 = -f(s)+s"1. This shows that f(-s) = -f(s). Proof of the theorem. We already know that the integral Z(sA;«) = JFAXO(x)^(x)|x|sAdjlx(x) is absolutely convergent if Re(s) > 1. Let Fi be a continuous function as in the lemma and define Fo by F0+F1 = 1. Thus 0 < Fj < 1 and Fo(x) = 0 if x < to and Fi(x) = 0 if x > ti. Take arbitrary B > 1. Then, for any a € R with a < B, we see that x a F 0 (x) < t o a " B x B . We define ZfafcQ)

= JFAxO(x)?i(x)|x|sAFj(|x|A)d|ix(x).

Then writing a = Re(s), we have JFAX

I *(x) I |x|° F o ( I x I A )d|i x (x) < t o a - B J FA x I ®(x) I |x£dji*(x).

Thus this integral is absolutely convergent for all s since B > 1. Thus Zo(s,A,;0) is an entire function of s. Now replacing X by X'1, s by l-s, O by O' and Fo by the function x h-» F^x" 1 ), by the same argument as above, we have an entire function of s defined on the whole s-plane, Z'od-s,*- 1 ;* 1 ) = JFAxO'(x)V1(x)|x|1A-sFi(|x|;1)d^ix(x). Let <£z(x) = O(zx) for z e FA X . Then the Fourier transform of O z is given by I z I A ' ^ ' Z - I a s a ^ rea( iy remarked. We now write, choosing a fundamental domain X of FAX/FX, F A X = U^F-{0}^X . Then

8.6. Functional equations of Hecke L-functions

A

269

Mx) |x|sAFj( |x|A)d|ix(x).

^

Similarly, we have

By the Poisson summation formula, we see that

Thus we see that x | AsFi(|x|A)djlx(x). By the variable change x H» X"1, the first integral equals Z'oQ-s,^"1;©') and thus

) I x | A-1-*(0)}X(x)|x|8AFi( I x | A)d|ix(x). Now we use the fact that FAX/FX = KxR + for a compact group K = F(A}/FX. Write X = Xfa as a product of characters of K and R+. Since log : R + = R and Hom cont (R,T) = R, ^ 0 0 = I x | A1 for some t e V-1R. Then we have

Thus we may assume that t = 0. Then ZiCs^OJ-Z'od-s.V1;®1) = JKWjixJR+{*l(O)-*(O)x}x8-2Fi(x)dx. Since K is compact, c(^) = jKX,d|LLx is a constant and vanishes if X * 1 on K. Anyway, by Lemma 1,

for a meromorphic function f(s) satisfying the following conditions: (i) f(s)-s"1 is entire and (ii) f(s)+f(-s) = 0. Since Z(s,X;O) = Z 0 (s,X;O)+Zi(s,^;©), the

270

8: Functional equations of Hecke L-functions

above fact shows the analytic continuation of Z(s,A,;O) because Z is entire. Moreover we have Z(s,?i;O) = Z0(s,?t;O) Since f is an odd function, we know the functional equation

because of the Fourier inversion formula !F.!F(O)(x) = O(-x). We have not proven the inversion formula yet but for the special function Oa, it is obvious by (3). We now give a sketch of a proof of the inversion formula when v is finite (a similar argument works also for R or C). Let f be a standard function and ^n( x ) = Oo(7Cnx) be the characteristic function of G5~nOv. Then we compute for G = Fv

We already know that ^(On)(x) = \K\ v"nOo(Ttr~nx) by (2). Instead of the above integral, we compute JGJGcX)n(y)f(x)(x,y)dja(x)(y,z)djl(y) = JGJGOn(y)(y,x+z)d^(y)f(x)d^(x) = JGJ(On)(x+z)f(x)d^l(x) = I G3 | v-nJGd>0(7Cr-nx)f(x-z)d^(x)

By taking the limit as n —> °o, we see that

f(-z) I 031 v"r = f(-z)J G O 0 (G3 r x)dn(x) = | G51 ^ because the difference of (I and JLL1 is I G5r I v1/2- This shows the result. Exercise 3. Show that Z^eF^K^) is absolutely convergent when F = Q. When X is the trivial character, we see that <E>x(x) = Of(xf)Ooo(Xeo) for the characteristic function Of of 6 and Ooo(Xoo) = exp(-7cZaGil XooG I 2 ). Thus 0(0) = 1 and by (4), J(O;0(0) = | D | - 1/2 O x (0) = | D | "1/2. Thus we have O'(0) = | D | ~1/2. By the proof of the theorem, we see that '(0)

(K =

8.6. Functional equations of Hecke L-functions

271

We now compute JKXd|Lix. By Corollary 1.1, we see that for K' where h(F) is the class number of F. It is then plain that K1 = X/E for X = {x e F x | N(x) = 1} where E is the subgroup of (f consisting of totally positive units. We then consider the logarithm map in (1.2.7) / : F x ^ R a . We decompose the Haar measure on C so that dxdy = rdrdG for r = I z 12 for z e C and the Haar measure d0 on T. Then we see easily that vol(/(X)//(E)) = J/(X)//(E)dr = | R | = R., where R = d e t ^ e ^ i j ) for a basis {ei,...,e s } (s = #(a)-l) of the torsionfree part of E. Note that the kernel Y of / can be written as ({±l}a(R)xTa(C))/M-(F) for the group |i(F) of roots of unity in F. Then we have J Y d0 = 2r(27c)Vw for w = #M<(F) (r = #(a(R)) and t = f ^ x 2r(27i)tR h(F) A u . Thus J K ?id|i x = — — — 0O — — , and we obtain Corollary 2 (Residue formula). We have 2r(27i)tRooh(F) r ^ Res s=1 C F (s)= . | D | 1 / 2

Chapter 9. Adelic Eisenstein series and Rankin products In this chapter, first we shall give an adelic interpretation of modular forms and then we will compute the Fourier expansion of adelic Eisenstein series for GL(2) over Q. After this computation, we will know that the Eisenstein series has analytic continuation with respect to the variable s. Using this fact, we will show the analytic continuability of the Rankin product and its functional equations.

§9.1. Modular forms on GL 2 (F A ) Hereafter, we assume that F is a totally real field. We start from the definition of modular forms on the group GL2(FA), which consists of invertible 2x2 matrices with coefficients in FA. We use the notation of Chapter 5. In particular, H denotes the upper half complex plane. We put, identifying F<x> with R1 for the setofrealembeddings I of F, G ^ = {x G GI^CFoo) = GL2CR)1 I det(x a ) > 0 for all o e I}. a . linear We put tions we x(z) (xo(z Then can= let theo)) group Ge*>+ act^ ond J(z) Z= =?{ ^ jvia fractional transformaO€ i, where

o+ = {x G G^ + | x(i) = i} for i = (^,

..., ^p[) e

Z.

Exercise 1. Show that for a e GL2(R) with det(a) > 0, a(i) = i (i = V—1) if and only if there exist t € R x and 0 G R such that fcosG -sinB^ a = V e cose} Es P eciall y C ^ = R X SO 2 (R). For a =

faa b^ A and z G #", we put j(a,z) = (cz+d). Then we see easily that

j(ap,z) = j(a,p(z))j(p,z). Thus if a ( 0 = p(/) = /, then j(ap,z) = j(a,0j(P,0 and the map a h ^ j ( a , 0 is a group homomorphism of RXSO2(R) into CX. In fcosB sinB^ ;A v fact, j(a,0 = tezo if a = t^s]nQ C Q S 6 J. We define jk(x,z) = UoK^c^o) for each positive integer k, x e Geo+ and ze Z, and | j(x,z) | s = Flo I K^c^zc) I s for s G C. Then the map x h-> jk(x,i) is a group homomorphism of CL+ into

^

J

cx. We consider the open compact subgroup S = G L 2 ( 6 ) = n p GL 2 (O ; ,) of GL2(FA) and its subgroup of finite index, for each integral ideal m

9.1. Modular forms on GL2(FA)

273

For a given character % : Cl(m) = ¥\X/Fx\](m)¥oo+ -^ T, a continuous function f : GL2(FA) —» C is called an adelic modular form (in a weak sense) of weight k, of character % and of level m if it satisfies the following condition: (Ml)

f(ocxu) = XmOOfWjkOw)"1 for all u e S(m)Coo+ and oce GL 2 (F),

fa b\ where %m( I) = IIV | m%v(dv)- Modular forms play the role of Hecke characters for the group GL(2) in place of GL(1). In fact, any Hecke character X : FA X = G L I ( F A ) -» C x for a general number field F satisfies the condition A,(ocxu) = 9i(x)uoo"^ for a e F x = GLi(F) and u e U(/rc)Foo+ for its infinity type £, which is an analog of (Ml). We can define, from the modular form f and a given element t e GL2(FAf) as above, a function ft: Z —» C as follows: for z = x+iy e Z (x, y € Foo), (y x we pick one element Uoo G Goo+ so that u^i) = z, for example, Uoo = L satisfies the requirement. This in particular shows that Goo+ = B00+C00+, where

We put ft(z) = f(tUoo)jk(Uoo,i). This definition of ft(z) does not depend on the choice of Uoo. In fact, if ueo(/) = u'oo(0> then uOo"1u'Oo(0 = / and hence c = Uoo^u'eo G Ceo+. That is, UooC = u'oo and

f(tu'oo)jk(u'oo,i) = f(tUooC)jk(UooC,i) = f(tuOo)jk(c,i)"1jk(c,i)jk(Uoo,i) = f t (z).

Now put r t = GL2(F)nt"1S(N)tGoo+. Then Tt is a subgroup of GL2(F), and if t = 1, (a. b^

1

r i = {a = I d j 6 GL 2 (O) \ ce m and det(a) e Foo+}. In particular, if F = Q, then fa b F i = T0(m) = {y = ^ J G SL 2 (Z) | c G m}. We now show that ft satisfies ft(y(z)) = %*(d)ft(z)jk(y,z) for y e Tt if t is of fa O^j the form , where %* is the ideal character associated to the idele character %. Thus ft is a modular form on Z for the discrete subgroup F t in the classical sense: if y G F t , then

274

9: Adelic Eisenstein series and Rankin products ft(y(z)) =

since YooUoo(i) = Y(z). The adele matrix Yoo has non-trivial components only at infinity and t is concentrated on finite places. Thus Ttt'Vf^t = YYf~!t = Y~t = tY~. Moreover t^yfhe S(m) and we know that f t (Y(z)) = f(tYooUoo)jk(Y-Uoo,i) = f(Ytt" V ^ = X m (d)- 1 f(tu oo )j k (Yoo,z)j k (Uoo,i) = Xm

Note that 1 = %(d) = %m(d)%*((d)) for the ideal character %* associated to the idele character %. Thus XmCd)"1 = %*(d). Similar computation shows that for more general a e GL2(F) with det(a) » 0 (the symbol "»" means that det(oc) is totally positive, i.e. det(a) a > 0 for all a e I), we have (la)

ft(a(z))j k (a,z)- 1 = f a r i t (z).

Thus to one modular form f on GL2(FA), we can attach a system of classical modular forms {ft}. This system is basically parametrized by the double coset space: GL2(F)\GL2(FA)/S(/n)Goo+ = GL2(F)\GL2(FAf)/S(/rc) which is an analog of the class group Cl(m) = Fx\FAx/U(fft)Foo+. In fact, it is known that if {ai}i=i h is a representative set of Cl(l) in FAX, then (lb)

GL 2 (F A ) = UiL1GL2(F)

ISin^G^ (approximation theorem).

Thus each modular form on GL2(FA) corresponds to a set of h classical modular forms on Z. We now impose the following holomorphy condition on f: (M2)

ft is a holomorphic function on Z for all t e GL2(FAf).

Let us give a proof of (lb): Consider Lf= 6 2 and L = O 2 as lattices in the column vector space V = F 2 . For each x e GL 2 (F Af ), xL = xLfflV is a lattice of V. We first show that there is a vector 0 * y e xL such that xL/Qy is torsion-free. Take one non-trivial vector y € xL. The places v such that the image y(v) of y in xL//vxL is zero are finitely many (note that y(v) = 0 if and only if xL/Oy has non-trivial py-torsion). Let Z be the set of such places. We choose z e xL so that the image z(v) in xL/pvxL is non-zero for every v e Z. We may assume that z and y are linearly independent over F. Then the images of z and y in xL/pvxL are linearly independent for almost all places v. Let Y be the finite set of places v where the images of z and y in xL//vxL are not linearly independent. Thus we can write z(v) = ?c(v)y(v) for v € Y with

9.1. Modular forms on GL2(FA)

275

X(v) e Ov/py. Since 0v//v has at least two elements, we can find be O such that (b modpy) * X(v) for all v e Y (by Chinese remainder theorem). Then put t = z-by. Then if v e Y, then t(v) = t mod pvxL * 0 by the choice of b. If v is in Z, then t(v) = z(v) ^ 0 by our choice of z. If v is in neither Z nor Y, then z and y are linearly independent in xL/p^xL and hence t(v) * 0. Thus xL/Ot is without torsion. Thus we may assume that xL/Oy is torsion-free from the first. Since xL/Oy can be embedded into F, it is isomorphic to an ideal a of O, which is protective because O is a Dedekind domain ([Bourl, VII.4.10]). Thus xL = at0py for an ideal a and t e xL. Take a e F Af such that aO = a, and write a = pai with p in F and for some i (we can choose the parity of P arbitrarily at infinite places by changing i if necessary). Then, writing the matrix for t = [tl 1 and y =

\h)

We may further assume that det(oc) » 0 by choosing a suitable parity of p. Thus fa- 0^ , x = ocI ju for u e S since S = {xe GL 2 (F A f ) | xL f = L f }. This shows (1) for m=O. For general m, we can easily approximate u as above by y e b^SbflGL^F) for b = k j

I so that yu e S(m). This is a special

case of the strong approximation theorem but it is not so hard to verify it in this case. When F = Q, we have already proven the strong approximation theorem as Lemma 6.1.1 and its proof applies to the general case. Now we define the Fourier expansion of modular forms. Since any modular form f on GL2(FA) has a prearranged move under the left translation by GL2(F) and the right translation by S(m)Coo+, by (Ml), it is determined by the value on the subgroup B(F A ) + = ( b e B(F A ) | det(boo) » 0}, where for any Q-algebra A, we put

B(A) = j f j j l l a e (A(g>QF)x and b e A(g>QF

Let f be a modular form as in (1). We write simply f(y,x) = f

for u e F A , we consider a unipotent matrix a(u)y

i]

=

o

X U

1 J*

thus f(y,x+£) = f <x(u)

In

Particular'

if

oc(u) =

(fy x\\ 0

1

(I u\

I . Now

.

Then

^ G F, then x(^) e GL 2 (F) and

= f(y,x). Thus f(y,x) is translation invariant

under F in the variable x. Thus for a fixed y, we can consider x h-» f(y,x) as a

276

9: Adelic Eisenstein series and Rankin products

continuous function on FA/F (which is of C°°-class in x«o G Foo). The group FA/F is a compact additive group. Thus we can expand f(y,x) into an adelic Fourier expansion on FA/F (see §8.4); namely,

where <>| runs over all additive characters <|) G Homcont(FA/F,T) and c(y,<()) = JFA/Ff(y,x)(|)(-x)d|i(x). Here the additive Haar measure \1 is normalized so that JI(FA/F) = 1. We already know that Homcont(FA/F,T) = F so that each character of FA/F is given by X H e(J;x) for the standard character e : FA/F —» T. Thus we can write this expansion as ^

for x e F A .

Let us verify several properties of this expansion. Since f is invariant under right multiplication by the matrix a = L f(uy,x) = f (2)

ffy

with u G U, we see that

x^i ^ a = f(y,x). This shows that c(uy,£) = c(y,£). Thus c(y,^) depends only on the ideal yO and yoo.

Similarly f(T|y, rjx) = f(y,x) for T[ e F x , and thus (3)

cCny,$) = c(y,^Ti) for ^ E F, 0 * r| G F.

Let ft = dO (de F Af x ) be the different of F/Q. Thus d'1 O is the maximal additive subgroup in Af of the form x 6 so that ef(x 6 ) = 1. Now for ue

6,

oc(u) =

G S(w), we know that

Therefore c(y,^) = c(y,^)e(^uy) for all u e c(y,^) ^ 0 , ^y G d"1 6 ; in other words, (4)

6.

f(y,x+uy) = f(y,x). This implies that if

£dyO is an integral ideal if c(y,£) * 0.

Now we compute the function

fa (A ft : Z -» C for t = L with a fixed

a G Af. Write a = aO for the corresponding ideal. Then for z = Xoo+VoJ e Z, taking

as Ueo, we see that

9.1. Modular forms on GL2(FA) (5a)

277

ft(z) = f(ay«,x«) = X ^ e

Especially, if ft is a holomorphic function, then -r^- = -—+ *\— kills each term and hence -—c(ay <*>,!;) = -2rci;ac( ay «>,!;). Thus c(ayoo,£) is a constant We

multiple of exp(-2rcTr(S;yoo)). We write this constant as c(^da); thus c(ayoo,£) = c(^da)exp(-2jcTr(^yoo)). We now show that c(^da) = 0 unless £ a > 0 for all a when F ^ Q . For that we pick one £ * 0 which is not totally positive. We then define [£] = {a e 11 ^ a < 0}, which is not empty. Fix one element a € [£] and take a totally positive unit e such that e° > 1 for a and 6T < 1 for all % ^ o. We can always find such a unit (see the proof of Dirichlet's unit theorem, Theorem 1.2.3). Then c(^da) = JFA/Ff(l,x)\|/(-^x)d|i(x)exp(27cTrF/Q©)

JF A /F

I f(l,x) I d^(x)exp(27iTrF/Q(^en)).

Note that £ a e o n -> -©© as n - » ©o and JjV* 1 -» 0 as n - » o © . Thus we know that exp(27cTrp/Q(^e n )) —> 0 as n —> ©o. This in particular shows that c(^da) = 0 unless £ a > 0 for all a e l .

When F = Q , this argument does

not work and we need to impose the following condition:

(M3) ft has Fourier expansion of the following form:

) f°ral1

tG

where n runs over a lattice of Q. As for the constant term ao = ao(ft), if we (y A restrict t to be (ye FAfx, x e FA), it is a function of y, and hence we fa

write ao(y). Since f is invariant under left multiplication by I 0 fu and under right multiplication by L

0\

0°)

I (a e F x )

for u e O x , the function y h-» ao(y)

factors through FAfx/Fx 6 x , which is the absolute class group. Thus we can define new functions n h-> a(n;f) by a(n;f) = c(^da) if n = ^da for \ e F + and ni->ao(rt;f) by ao(n;f) = ao(yd"!) if rt=yO. We denote by for the space of functions satisfying the conditions (Ml-3). Then f e has adelic Fourier expansion of the following type:

278 (5b)

9: Adelic Eisenstein series and Rankin products f(y,x) =

where ni-» a(n;f) e C is a function of fractional ideals vanishing outside integral ideals and n h-> ao(n;f) factors through the ideal class group of F. The space of cusp forms Sk(w,%) is the subspace of Mk(m,%) consisting of functions f e Mk(m,%) satisfying the following cuspidal condition for the standard Haar measure dji on FA/F:

(S)

JFA/Ff(jo i] x Jd|i(u) = 0 for all x e GL2(FA).

This just means the vanishing of the constant term of ft for all t. Since ft I a = fotrit for a e GL2(F) with det(a) e F+, this simply implies that ft is a cusp form for all t. When F = Q, we know from (lb) that GL2(A) = GL2(Q)S(m)Goo+ since (\ (A A x = QXU(1)R+X (8.1.3a). We already know that ft for t = L A gives a homomorphic modular form in #4(ro(ffO>%*) since Cl(m) = (Z/m)x and the character %* can be considered as a Dirichlet character. Conversely, starting from <|) e ^k(ro(^),%*) and writing each X E GL 2 (A) as a u with as G L 2 ( Q ) and u € S(m)Goo+, we define f : G L 2 ( A ) - ^ C by x f( ) = Xm(u)
Thus f(x) is well defined independently of the choice of a and u. It is obvious from (5.1.la,b) that f e Mk(m,%) and fi = 0. Thus (6)

Mk(m,%) = fA4(r0(m),%*) and 5 k ( « , x ) s 4 ( T o W , X * ) via f H> fi.

In this sense, we can take Mk(m,%) as a generalization of the space for general totally real field F. We have, for t = L

A and

fe

9.1. Modular forms on GL2(FA) (7)

ft(z) = Xo«5Ea-U-i c^a)txp(2n^TT(^z))

279 for c(a) e C.

In particular, when F = Q and a = 1, the above expansion has the usual form: X°° c(n)exp(27cV-Inz). In general, c(y£) = c ( y U ) = c(^dyO)exp(-27iTr(^y)). Thus we may regard the Fourier coefficient c(a) as a function of integral ideals a n c(a). The L-function of f is then defined by L(s,f) = £ac(a)N(a)"s. This function is known to have an analytic continuation and a functional equation similar to Hecke L-functions. These L-functions are first investigated in detail by Hecke and have now become very important tools in number theory. Of course, when F = Q, this L-function is nothing but the one studied in §5.5. In this book, however, we do not go into details of the theory of such L-functions. We list [JL], [G] and [W3] as standard textbooks for this subject. Instead, we introduce now the Eisenstein series as an example of these modular forms and try to give some account of how to compute the Fourier coefficients of Eisenstein series when F = Q in the following section. Let T = GL2(F)rif1U(m)tGoo+ for the above t. Thus r = {r

d]

e GL 2 (F) I a, d e O, c e am, d e a'1, ad-bc e E } ,

where E is the group of all totally positive units in (?. Let

Then

j(y,z) k = (N(d)/\ N(d)\ )k if y = f* Jl e r . . .

Thus we see

j(Y5,z) k =j(y,8(z)) k j(5,z) k = (iV(d)/|iV(d)|)kj(5,z)k for y e r_. Now we put for a character % : (O/m)x -> T such that %(e) = (N(e)/ \ N(e) I ) k

E r (z,s,%) = y s X 7 er eo \r%( 8 )J( 5 ' z )" k I J(5>z) I "2s>

(8) ffSL

where

%\\

b\\ A

= %(d).

Then this series is absolutely convergent if

Re(s) > l-(k/2) (see §§2.5 and 2.8, where we expressed Eisenstein series in terms of Shintani zeta functions) and Er(y(z)) = %(y)"1Er(z)j(y,z)k for y e T by definition. We now interpret this definition in adelic language. In the adelic case, the object corresponding to the congruence subgroup T is GL2(F), and that of Too is Boo = 0*B(F), where

280

9: Adelic Eisenstein series and Rankin products

B(F) = {(* J] I a e F x and b e

F|.

Let % : Cl(m) = F A x /F x U(/rt)Foo+ -> T be a character and suppose that %(Xoo) = (N(Xoo)/ |N(Xoo) I ) k . We define the following functions on GL 2 (F A ):

(9) ri(x) = < M y ' A i f x 10 otherwise

=

(o l j a u ^a

G FA><

and

UG

s w c

( ) ~+)>

and a

b

l l = jnvUXv(dv) if (* J ) e B(FA)S(m)Goo+,

^C

d

^

%#ff

10

otherwise.

These functions are well defined. To see this, let Lf = 6 2 (column vectors).

(y

b>

i

a

v\

Then, if au = a'u' as above, then looking at the second row of b> >y (y \ fy b \ aL uLf = a' u'Lf which only depends on a and a' because uLf = u'Lf, we know a d = a' 6 . Thus I af I A = I a'f I A- By taking the determinants of both sides, we then conclude that I yf I A = I y'f I A- O n the other hand, Im(Xoo(i)) = yoo = y'oo, which shows that I y I A = I yf I A- Similarly %# is well defined. By the product formula |£ I A = 1 for t, G F X (8.1.5), we know that Ti(yx)=r|(x) if y e B^. We also see that %#(yx)j(YXoo,i)"k = %#(x)j(Xoo,i)"k for all y e B^. Exercise 2. Suppose that %(xoo) = (Af(Xoo)/1 A^(Xoo) | ) k . Then show that X#(yx)j(yxeo,i)-k = x # (x)j(x oe ,i)- k for all ye B^. We define E*(x,s,%) = IYGBeo\GL2(F)X#(yx)rl(/Yx)sj-k(yxoo,i). Suppose that

xf = t = L

a

f

X (Yt)Ti(yt) * 0. Write y = with s =

A.

Then yt e B(FA)S(m)Goo+ ^

°

if and only if

*

a , then if % r|(yx) ^ 0, there exists s e S(m)

\° 11aJ

. . Thus c e a = sC sCl lOt d e O and ca+dO= O. This shows that

we can find e e O and f e a

such that cf-de = 1. Namely 8 =

One can easily show that this correspondence

J e I\

9.1. Modular forms on GL2(FA)

281

BooMylyte B(FA)S(m)Goo+} s y H-> 5 e TooXT is bijective and thus each term of the summation of Ep and E* can be naturally identified. Moreover by the above computation, writing (yt)f = bfSf as above for b e B(Ap), we see that Tj((yt)f) = | det(bfSf) I A = I det(yf) I A I a I A- On the other hand, TI((YX)OO) = Im(yxoo(i)) = |j(y,Xoo(i)) |"2|det(yoo) I ATI(XOO). Thus (10a)

ri(yt) = I j(y,xo.(i)) I "2 I det(Y-) I AIm(xoo(i)) | det(yf) | A | a | A = |a|AIm(x0O(i))lj(Y,x0O(i))|-2.

Thus we know that E*(txoe,s,%)j(xoo,i)k = | a | AsEr(xo.(i),s JC*' 1 ),

(10b)

because %*((d))%(dm) = %(d) = 1 for the ideal character %* associated with %. This shows the convergence and we now know E* corresponds to Ep naturally. By definition, we have E*(yx,s,%) = E*(x,s,%) for y e GL 2 (F) and E*(ux,s,%) = %(u)E*(x,s,%) for u e 6 X . Thus for z e F A f x , the modular form %-1(z) | z | AkE*(zx,s,%) depends only on the class of z modulo 6 XFX. Note that F Af x / 6 X F X = F A X / O xFxFooX which is isomorphic to the absolute ideal class group Cl of F. We now define (11)

Ek(x,s,%) = Ek,m(x,s,%) =L m (k+2s,%- 1 )£ zGC1 %- 1 (z) | z | AkE*(zx,s,%) and Gk(x,s,%) = Gk,N(x,s,%) = %(det(x))Ek(xxf,s,%-1),

where

%=

(0 -I) n

\ with a finite idele

m such that

mO = m and

^m(s,%"1) = Hn%l{n)N(n)~s in which the sum is taken over n prime to m. Note that for z1 e FA X , we have ^S"*

'V" ( v \

-7

A T**('77*T[ C f\f\ — ^V

7

M771

^

I P

77

A P*^7Y

<2 V^

= X(z 1 )lz'| A - k X zeC iX" 1 (z)E*(zx ) s ; %). Thus Ek satisfies (Ml) for %k : z \-> %{z) \ z | A"k. Using the fact that a b ^ _ , ( d -c) lmc dj i^-mb a we conclude easily that G k also satisfies (Ml) for %k. We now state the principal result:

282

9: Adelic Eisenstein series and Rankin products

Theorem 1 (Hecke-Shimura). Let % be a finite order Hecke character modulo m. Define a function on the set offractional ideals by am>5C(a) = Hg^J&ftNiGf1 and a'm^Ca) = E O a %(a/£W(£) m if a is integral and otherwise

om>x(a)

= o'm.xCfl) = 0. Here we understand %(a) = 0 if a and m have a non-trivial common factor. Let k be a positive integer such that %(xoo)=-j

iV(Xoo) k

^nr- Then

lA^()r Ek,m(x,s,%) and Gk,OT(x,s,%) c#« £e continued to meromorphic functions in s $0 that there exists a non-zero entire function f w s such that f(s)Gk(x,s,%) and f(s)Ek(x,s,%) are entire. Moreover they are finite at s = 0, and except when k = 2, F = Q, and % is trivial, we have

and if m* O,

l l,0, C anti C are non-zero constants depending on k and m. In the exceptional case when F = Q, % is trivial and k = 2, G2,m(x,0,id) is nonholomorphic. Here e(/^eoyoo)e(^Xoo) = exp(2jc/Tr(^z)) for z = x«o+iyoo and * = V-T. We will prove this theorem for F = Q in the following two sections. A proof for the general fields F can be found in [Sh9] and [H8, §6].

§9.2. Fourier expansion of Eisenstein series In this section, we assume that F = Q and compute the Fourier coefficients of Eisenstein series. I follow Shimura [Sh2, 7, 9] in this computation, which can be generalized to bigger groups (e.g. symplectic and unitary groups as was done in [Sh9]). We exploit the fact that our group is GL(2) to simplify the computation in many places. We restate the definition of the Eisenstein series when F = Q. Let N be a positive integer. We change the notation here and the ideal NZ plays the role of m in the previous section. We first prove Theorem 1.1 when N > 1. We will later remove this assumption to include the special case of N = 1. Let X : (Z/NZ) X = C1(N) -> C x be a Dirichlet character with %(-l) = (-l) k (0 < k G Z). We define two functions TI,%# : GL2(A) -> C by

9.2. Fourier expansion of Eisenstein series (y

ii

b>

283

i

I y I A if x = I Q J a u (u e ) otherwise,

xf#ffc l )l ={I n , X (d ) i f f! : i e a

D

M P

V

^

JJ

10

p

B(A)S(N)G«+,

otherwise.

Then the function x h-> %#T|(x)sj(Xoo,i)"k is invariant under left multiplication by elements in B = {±1

x

11 a e Q x , b e Q}. Then

E*(x,s,%) =

where GL 2 (Q) + = {a e GL 2 (Q) I det(a) > 0} and B + = BflGL 2 (Q) + . Put (0 - n e = L 0 I. We compute the Fourier expansion of E(x,s) = E*(x£f ,s,%) and later relate it to the Eisenstein series Gk(x,s,%) in the theorem. Note that E(yx,s) = E(x,s) for all y e GL 2 (Q), and hence E(x,s) has a Fourier expansion. We define its Fourier coefficients for £ e F by b(5,w,s) = JA/QE(a(x)w,s)e(-^x)d|Li(x), where w e A X B(A), JLL is the additive Haar measure on A/Q such that

(\ x]

|J,(A/Q) = 1 and oc(x) = I

. We observe that

a(x)w Suppose that w e AXB(A)+. Then the non-triviality of %#(ya(x)wef"1)r|(Ya(x)w£f-1)s means that ya(x)we{1 e B(A)S(N)Goo+ and ya(x)w e B(A)S(N)Goo+6f. fa b^| fa b\ Since u = 0 . mod N for all u e S(N), if we write y = I , I, then

c^O. Thus

Y=[

c2det(Y)

c

^JceaCc-M). That is, y e B^.QxeU for

U = {a(x) | X G Q). Thus B+\B+QxeU = Q + eU. Hence we see that

284

9: Adelic Eisenstein series and Rankin products

b($,w,s) Xx # (eya(* + 8)w£f 1)Ti(eya(x+5)wef-1)sj(yoo(a(x+8)w)e«,0"ke(^x)d|i.

= J

Q6

We now prove a lemma, which shows that the above summation with respect to y e Q is in fact reduced to the summation over a fractional ideal of Q. Lemma 1. Write w = a(x)a

(y 0^1 0

= a

(y x] n

for a G A . Then we have

(ewe'^f e B(Af)S(N) (i.e. %#(ewef'1) * 0) if and only if X G a 4 N Z , a e y" x Z and axZ+ayZ = Z. Proof. a

1

By computation, we see that 0\

(ewe )f =

(

a

0>

l

• From this,

. . . . . e B(A)S(N) implies ax G NZ, ay G Z and axZ+ayZ = Z

(<=> axZ+ayZ = Z). On the other hand, if a x e NZ, ay G Z and axZ+ayZ = Z, then we can find t,s G Z such that axt+ays = 1 and S u = fl^-ax

1

1 G S(N) and

ayj

(ewe'V 1 = f Y

(1)

which shows the converse. (y

0>

i Applying the above lemma to 7a(x+8) (instead of w itself) for a given y -1 G y Z, we see that %#(eYa(x+8)wef"1) * 0 <=> y(x+8) e NZ and y(x+8)Z+yyZ = Z. Let O = O 7 : Af -> C be the following function depending on y and y: O(x) = np^)p(Xp) and yy p Z p . Thus if NZ+yyZ = Z (i.e. yyZ is prime to N), then %#(eyx(x+8)wef'1) ^ 0 if and only if O(y(x+8)) = 1. Thus we see that

w,s) = J A/Q

J%#(eycc(x)we71)O(yx)r|(eya(x)wef"1)sj(yoo(a(x)w)oo,0"ke(^x)dja(x),

9.2. Fourier expansion of Eisenstein series

285

where y runs over y"1Zf|R+ such that yyZ+NZ = Z. By (1), we can compute %#(eya(x)wef"1)ri(eYa(x)wef"1) explicitly: Lemma 2. Suppose that O(yx) = 1 and yxZ+yyZ = Z for y e Q+ and y x\ and z = Xoo+V-lyoo e 9{

[

rj(£ya(x)we f - 1 ) = I yf2y I A I z I "2 and %#(eya(x)wef"1) = n Therefore, for %N(yy) = ri p | N X P (Y P y p )

\s) = X J

0
2s k

' I y f I A s j A O(yx)z- k I yoo I s I z I - 2s e(-^x)dja(x),

because %(Y^NyN^)%N(yy) = %(y) and %(y^ N V N) ) = %(yyZ) as the ideal character (where y(N)ITp | N y P = y). Now we compute the following two integrals: .1

i

r

and

Then of course, b($,(j j],s) = bK§,(j i],s)b-(§,^ *j,s). compute the finite part of the integral. (<=> yZ p = yp^Zp). Then

First suppose that

We first

yy p Z p = Zj

" 1 jNZ p e p(-^r 1 x)d|ip(x) = I y 4 N | p J Zp e p (^y- 1 Nx)d|ip(x) " y p N | p if £ y ! N e Z p (i.e. \ e N ' V p ' ^ p ) , 0 otherwise. Next suppose that pZ p 3 yy p Z p (<=> yv'l7uv 3 yp^Zp) (in particular, p is prime to N). Then we have

286

9: Adelic Eisenstein series and Rankin products

if x h-» epC^y^x) is non-trivial on pZ p , i.e. ttf1 £ p^Zp. Thus we may assume that § G yp^Zp. Then

Thus we know that if £ e y^N^Z, then

J.s) = x(y) Xx = Z(y) I yf I A - ^ N " 1

X%"' (^Z) I fyy)f I

0<7ey~1Z

To compute this sum, we introduce the Mobius function \i on the set of ideals with values in {±1,0}. Let X : Z -» C be any Dirichlet character and consider the L-function

Then write L(SjX)"1 in the form of a Dirichlet series Z°° |i.(n)X(n)n"s. Then n=l

and hence |i(l) = 1, |j,(p) = -1 for a prime p and |x(piP2 Pr) = (~l)r for distinct primes pj and |i(n) = 0 if n has a square factor. For any positive integer m, by the above definition, we see that

In particular, if we specialize s = 0, then we see that rl if m = 1,

2.o
if

m

> 1.

Thus for positive integers m and n,

We want to interchange the two summations: for each divisor d of m, the r with d I (r,m) is just the multiples of d; thus such an r runs over the set {d, 2d, ..., (m/d)d). Thus

9.2. Fourier expansion of Eisenstein series

287 ,

exp(-27ujdn/m)

exp(-2jc/jn/d) by d i-» m/d. Note that by the orthogonality relation of characters of Z/dZ v^d , „ .. /1 exp(-27izjn/d) = i . J-1 Id if d I n . This shows that From this fact, we conclude that

= X(y) I Yf I A " 8 - ^ ^ - 1

X X " 1 CyyZ) I (7V)f I A 2 s + k I r e ( Z / y y Z ) xexp(-2 7 ci^Ny/ 7 y) 0
= x(y) I yf I A - ^ N - ^ ^ z x k Y y Z ) I eyy)f I A2s+kIo
Zn=15C (")n

5>(n/d)d =

= L(2s+k)x"1)-1Xo
JL(2s + k,x- 1 )- 1 Xo x' 1 )' 1 L(2s+k-l,x" 1 ) if % = 0.

We now know that (2)

1

{

k

-

if

U(2s+k-l,x-1) if \ = 0. Now we compute y"sbo.(^,[0 A,s) = J_°° (x + /y)"k Ix+/y I "2sexp(-2ra^x)dx. Consider the function

C,(z;a,$) = j°°e'^ix+l^x^dx

Re(z) > 0 and ct,p e C. Here for z e C

x

a

for z e C with

z = | z! a e ' a 9

writing

288

9: Adelic Eisenstein series and Rankin products

z = I z I e*e with -K < 0 < %. Then one knows that this integral is convergent if Re((3) > 0 by a standard estimate. The divergence of the integral when Re(s) < 0 is caused by the singularity at 0 of the integrand. Thus by converting the integral into a contour integral on for sufficiently small e > 0, we can avoid the singularity at 0 and have an integral expression for (e 2mP -l)£(z;a,P), which is convergent for all p e C: +o

°

(e27Ifp-l)C(z;a,p) = z*\^zfi.+zh)*'l$'lGl&l

(see §2.2).

Thus the function (e 2 7 u P -l)£(z;a,p) is holomorphic on the whole space H ' x C x C , where H' = {z e C | Re(z) > 0}. By using the well known formula: T(p)r(l-p) = 2ni(en®-tn®y\ we know that co(z;a,p) = ZPr(p)"1C(z;a,p) = ^ - r is also holomorphic on H' x C 2 . When P = 0, by computing the residue of (l+z'hy^h^t'1 at t = 0 (see the computation in Exercise 2.2.1), we know that co(z;a,O) = 1. On the other hand, when a = 1,

Jp^a+z-HrHP-V^t = jmt*'ltldt

= (e27l/p-l)r(p) (see (2.2.2)).

Thus we again obtain co(z;l,p) = l by rflJ)r(l-p) = 27ci(em'p-e"w*)-1. Actually we will prove the following functional equation in Lemma 3.1 (see [Sh6]): co(z;l-p,l-a) = co(z;a,p); and the above evaluation follows if one knows either co(z;l,p) = l or co(z;a,O) = 1. Lemma 3. We have

rrk(27c)k+T(k+s)-1(2y)-s^k+s"1e-27C^co(47T^y;k+s,s) if § > 0, = jr k (27i)T(s)- 1 (2y)- s - k Ul s - 1 e- 27c| ^ ly co(4TcUly;s,k+s) if § < 0, Lr k (27c) k+2 T(k+s)- 1 r(s)- 1 r(k+2s-l)(47cy) 1 - k - 2s if § = 0. We will give a proof of this lemma later. By admitting this lemma, we now write down, for T c (s) = (27c)"T(s),

9.2. Fourier expansion of Eisenstein series (3a)

289

L(2s+k,x)%(y)E*(^

N~ s ~ k

v^.. n=1

+(21 y I A ) -

k

. i-s-k+i

| ( ny)flA )

,

„ , Jtiz Jtiz+(nx i

^2 s( kn+ iy, x z( y)) (e ( ^

^ I M f f ^-2s-k+i,z(nyZ)e(-

^

n=1

where as>%(yZ) = ZdZz.yZX(d)ds and z = Xoo+/yoo. Let x = I

Q

I. Then

we see from x P l f ) = J t P W ' 1 = (-l) k , E*(zx) = %(z) | z | A'kE*(x) and

that

(3b)

G k ([o ij,s,x) = (-D k N k L(2s+k,x)%(y)E*((-N- 1 ) f ^ jjxf.s.x" 1 ) NXfYyoo

AJ

N?

r(k+s)r(s)

oo

+rc(s+k)-1 Xl(ny)f \l'k+l a^.k+ia^y 2 )^ 1 1 2 ^ 1 1 *)^©^^ n=l

+r c (s)- 1 (21 y I A ) ' k a.2s.k+i,x(nyz)e(-nz-(nx)f)co(47inyeo;s,k+s)}. The explicit Fourier expansion (3a,b) shows the analytic continuability of Ek and Gk because we already know the meromorphy of L(s,%) and by an estimate similar to (5.1.4a,b), we easily know the convergence of the Fourier expansion for all s e C as long as the constant term is finite. Writing mZ (0 < m e Z) for nyZ, we see that |(ny) f | 1 A k - s a. 2s . k+1 , x (nyZ) | s = 0 = mk-1

I

k1

m/dh*d

X

Noting that {r(2s)/r(s)}| s = 0 = 2-1 and Ress=0L(2s+l,%) =

^^Yl^d-ip-1),

we now know from the above formula and (3b) that for C = V ^ P c k - l ) ! " 1 ^ ) * and for U Z = I Q

*

0 < k e Z,

290

9: Adelic Eisenstein series and Rankin products

(4a) Gk(uz,0,%) = c | ^ % ^ -

8

^

and for C = V ^ (4b)

Gk(u2,l-k,X) = C J L L ( 1 2 k > X ) + f> k _ u (n)e(nz)|, I n=l J

where 8k,j is the Kronecker symbol and 8x,id = 1 or 0 according as % = id or not, cp(L) = #((Z/LZ)X) and LL(s,%) = {n p | L (l-X(p)p" s )}^(s,X). Now we assume that N = 1 and hence % = id. The only difference in computation from the case when N > 1 is that for w e A X B(A) + , the non-triviality of %#(ya(x)wef"1)r|(Ya(x)wef"1)s does not necessarily mean that (a b\ fc"2det(y) c"!a^ , c * 0 for Y = AV Since we have y = ^ cex(c d) if rt \ c aJ \ 0 1y c ^ 0, we have GL2(Q)+ = B+Q^B^-Q^U (disjoint). Thus we have an extra factor coming from B + Q x in the computation of

5

J

Note that, by the formula IAJQ&\L = 1 (see §8.6), s

s

s

J A / Q 1 ! (uz) \[/(~^x)diLL = I y I A J A /Q\|/(-^x)dji = 8 ?>0 1 y I A .

Thus we have, for Gs(n) = as,id(n) (5a) GuCu.s.id) = C(k+2s) I y I A S + ^ { 2 K ( 2 I y I ^ n

r(k+s)F(s)

00

+rc(s+k)-1^|(ny)f|^s~ka.2s.k+i(nyZ)e(nz)e((nx)f)co(47tny»;k+s)s) n=l

+r c (s)" 1 (21 y I A )- k £|(ny) f f A ~ s a. 2s . k+1 (nyZ)e(-nz)e(-(nx) f )co(4jcny 00 ;s > k+s)} ( n=l

and by the functional equation for the Riemann zeta function, we have for C = V=lk(k-l)!-1(27c)k and C = 4-ik2kn (5b)

C-1Gk,i(x,0,id) = 2-1C(l-k)+8u(2jc I y I AY^^I

^k-

9.2. Fourier expansion of Eisenstein series

291

This finishes the proof of the theorem. We note here a byproduct of our computation: Theorem 1. Let the notation be as above. Then Fc(s+k)Gk,N(x,s,%) is entire if Re(s) > - 1 — and %Nk is non-trivial. When % is trivial and k = 0, it is 2 —k

entire if Re(s) > —-—. The singularities of this function are at most simple poles. We now prove Lemma 3. Following the argument given in [Sh6], we show that (6)

b(^y;a,p) = f

(x4-/y)- a (x~/y)- p exp(-2^x)dx ,p) if £ > 0.

The case where £ < 0 can be dealt with similarly. Then the lemma is the special case where a = k+s and p = s. We have

Here note that a = y-/x e Hf (i.e. Re(a) > 0). Recall the formula in Exercise 2.4.2: Jo~ e ^ V ^ d u = a"T(s) if a e H' and Re(s) > 0. Thus (y-/x)- a = r(a)- 1 J 0 ° O exp(-(y-/x)u)u a - 1 du if Re(a) > 0. Using this formula, we obtain (7) b(^,y;a,p) = /p

exp(-uy)ua-1 Now by putting f(x) = (y+/x)"p, we consider the inner integral JP

-°°

exp(zx(u-2^))(y+/x)"pdx= Jf°° exp(27i/x H: ^)(y+/x)- p dx -°° 2TC

^()(

2K

Here jT(f) denotes the Fourier transform of f. On the other hand, we also know that

292

9: Adelic Eisenstein series and Rankin products

Now define a function g : R —> C by fexp(-yu)u p ' 1 if u > 0, g(u) = i [0 if u < 0. Then, by (*), we see that f(x) = (y+ix)-P = r ( p ) ' 1 Jf exp(-27c/x^-)g(u)du = (p) J ( g ) ( ^ ) ° 2K 2% Thus J(f)(x) = 27cr(p)"1 JJ(g)(27tx) = 27tr(P)-1g(-27tx) by the Fourier inversion formula. Thus we obtain f- ixC-2,*), ,BJ j2 I cr(p)- 1 exp(-y(u-2^))(u-2^)P- 1 if u > 2 ^ , I e v ^J (y+zx)"pdx = s [0 otherwise. We plug this in (3) and then obtain b(^,y;a,p) = /P-(X(27r)r(a)-1r(p)-1 J^exp(-2yv)(v+^)a-1(v-7U^)P-1dv. By the simple variable change (V-TC^)/2TC^ h-> t, we obtain the desired formula.

§9.3.

Functional equation for Eisenstein series

Hereafter we always assume F = Q. Assuming that % is primitive modulo N (allowing the case % = id and N = 1), we now look at the explicit Fourier expansion of Gk(x,s,%) given in (2.3b) to know whether there is a functional equation for Eisenstein series Ek and Gk. We know from (1.9) that s,%) for z = x ee (0, where T = T0(N) and Er(z,s,%) = ysS7€reeNr%*-1(5)j(8,z)-k I j(8,z) | "2s. Note f a b>| that roo\T = R/{±l} via L N d I h-> (cN,d) for R = {(cN,d) e Z 2 | cNZ+dZ = Z } . Thus, regarding %* as a Dirichlet character on (Z/NZ)X, Er(z,s,%) = 2- 1 y s X(cN,d)6R^*" 1 ( d )( cNz+d )" k I c N z + d I "2s* Note that RxN = {(mN,n) I (m,n) e Z 2 } via (cN,d)xt h-> (tcN.td). Thus E'k,N(z,s,%) = y s X (mNtnM0>0 )%*" 1 (n)(mNz + n)- k | mNz+n | "2s

t=i

= 2L(2s+k,%-- 11)Er(z,s,%) = 2Ek,N(Xco,s,%)j(xoo,0k.

9.3. Functional equation of Eisenstein series Let x = L .

n

293

I. Then %l = -N'lz and we have

Efk,N(z,s,%) I kx =

N^E'N^SOCXNZ)*

= 2Nk-1Ek,N(Teoxee,s,x)j(xee,0k

= 2Nk-1Ek,N(Txooxf-1,s,%)j(xeo,0k = 2Nk-1Ek,N(XooXf-1,s,%)j(Xoo,0k = 2Nk-1x(-Nf-1)Ek,N(xOoXf,s,%)j(xee,0k = 2N-1(-l)kGk,N(Xoo,s,%)J(Xoo,0k. Now we have, for T c (s) =

(1)

(2TC)'T(S),

r c (s+k)r c (s)G k (^ *},s,x) = 8%,idrc(s+k)rc(s)C(k+2s) | y | As +*k(2N)"s{(2 | y | A)1-k-Tc(k+2s-l)L(2s+k-l,x) +r c (s) 2J( n y)f | A

a-2s-k+i,x(nyZ)e(nz+(nx)f)co(47cnyoo;k+s,s)

n=l

+Fc(s+k)(21 y I A)"k ^|(ny)f | A s(T.2s-k+i,%(nyZ)e(-n z-(nx)f)co(47inyoo;s: n=l

The idea is to compute the Fourier expansion of E k directly and relate it to Gk(x,l-k-s,%). Here we assume that % is a primitive character modulo N allowing the identity character when N = l . We know, for z = x«o(0, I mNz+n | "2s. Since we have shown in the previous section how to compute the adelic Fourier expansion of Gk, here we shall give a computation in a classical way:

E k (x-,s,)c) = 2- 1 y s X( mN ,nM0,0)Z Jk ' 1 ( n )( mNz+n )" k I m N z + n I "* (mz + - - + n ) ' k ' **

be(Z/NZ)

x

= ysL(k+2s,x*-1)+N"k"2s

m=l n=-oo

^

X5C*"1 (b) £ be(Z/NZ)x

S(mz+^;k+s,s),

m=l

where S(z;a,p) = Sm€z(z+m)"a(z+m)"p (a,p e C). For each z e C a n d s e C, we define zs = | z | s e t s 9 writing z = | z | e / e with -rc < G < n. We compute the Fourier transform of cp(y;a,P;x) = (x+/y)"a(x-iy)'P. Note that S(z;a,P) = 9(y;a,P;x+m) for z = x+/y. We have by the Poisson summation formula (Theorem 8.4.1) S(z;oc,p) = ]T nGZ e(nx)B(y,n;a,p)

294

9: Adelic Eisenstein series and Rankin products

where

B(y,£;oc,p) = £

e(-£x)(p(y;oc,p;x)dx.

Exercise 1. To justify the use of the Poisson summation formula, show: (1) If Re(oc+p) 2. 1, 9(y;a,(3;x) is continuous and bounded as a function of x and is integrable; (ii) XneZ I B(y,n+x;a,(3) | is convergent locally uniformly in x. (You may use Lemma 2.3 and Lemma 2 below.) By Lemma 2.3, we have, for z = x+/y e M

Ek((jJ *],s,%) = ysL(k+2s,%*-1)

(2)

+N-k-2s

£ % *-i ( b ) £ £ e ( m n x + ^)B(my,n;k+s,s) b€(Z/NZ)x

s

1

W

m=ln€Z

k 2s k

= y L(k+2s,%*" )+N- - /' (27i;)

s

x{(27c)k+T(k+s)-1r(s)-1r(k+2s-l)(47cy)1-k-s u

X 1

n>0b€(Z/NZ)x

'

f

n>0be(Z/NZ)x

= y*L(k+2s,X-V2-sN-^f*5x

Xl

be(Z/NZ)x k+s-1

°° m=l

m=l

^%*~l

I2

(b) ^m 1 " 1 "" 2 8 m=l

m

m

id{r(2s+k-l)L(2s+k-l,X)2S

rc(k+s)rc(s)

Here we have used Lemma 2.3.2 to obtain the last equality. Now we replace s in (1) by 1-k-s and compute for z = x+iy s 9{

r c (l-s)r c (l-k-s)Gk([j i],l-k-s,z) = 8z,idrc(l-s)rc(l-k-s)C(2-k-2s) | y | A s +i k N s+k - 1 2 k+s - 1 {(2y)T c (l-k-2s)L(2s+k-l,%) + r c ( 1-k-s) ]T ~=1 n" s ak+2s-i,z(n)e(nz)co(4jcny; 1 -s, 1 -k-s) +r c (l-s)(2y)- k X~ =1 n" s " k o k+ 2 S -i, z (nyZ)e(-nz)o)(45my«;l-k-s,l-s)}. We will prove the functional equation co(z;l-p,l-oc) = co(z;a,P) later. Then, noting the facts that F(s+1) = sF(s) and %(-l) = (-l) k , we know that r k (2N) 1 - s - k r c (l-s)G k , N (w ) l-k-s ) x)-2 s N k+2s J k r c (s+k)G(x- 1 )- 1 E k , N (w,s ) x) = c(s)y s +d(s)y 1 - k - s

9.3. Functional equation of Eisenstein series

295

for suitable meromorphic functions c(s) and d(s) of s. Since we know that Gk and Ek are both modular forms of weight k and of character %. Then we see for c

d1

e r o (N) that c(s)ys | cz+d | -2s-k+d(s)y1-k-s I cz+d 12s+k"2 = (c(s)ys+d(s)y1-k-s) I ky = X(Y)(c(s)ys+d(s)y1-k-s).

By choosing y and Y = [ c.

d.

I e Fo(N) suitably, we can make

- x(y) \d z+df

»oforatalostall,

This implies c(s) = d(s) = 0 for all s. Since GL2(A) = GL2(Q)S(N)Goo+, the above identity between Ek and Gk holds all over GL2(A). That is, we have Theorem 1. Suppose that % is a primitive character modulo N (we allow the trivial character when N = 1). Then we have

Exercise 2. Show the functional equation for L(s,%) using Theorem 1. We need to prove Lemma 1. We have

co(z;l-(},l-a) = co(z;a,p).

Proof. By definition, £(z;a,p) = J°°e~zx ( x + i y ^ x ^ d x , which converges absolutely for arbitrary a e C if Re(p) > 0 and Re(z) > 0. We compute

irx^dx = JVZX JV^V-Mux^dx, because

r(oc)(x+l)- a = J o °°e" u(x+1) u a - 1 du if Re(a) > 0 (Exercise 2.4.2).

From Fubini's theorem, we see that

-Mxdu = r(p) = r(p)za-P JV^u^cu+iyPdu = r(p)za-PC(z;i-p,a). This implies the functional equation of co because co(z;a,p) = Now we prove the following estimate for our later use:

296

9: Adelic Eisenstein series and Rankin products

Lemma 2. For any given compact subset T in C 2 , we can find two positive real numbers A and B such that if (oc,p) e T and y e R+, then |co(y;oc,p)| Proof. We have C(y;cc,p+1) = J ^ e ' ^ C x + l f ' ^ d x . Using the formulas e yX

" = ^(-y" l e " y X )' ^ { ( x + i r ' x P ) = ( we perform the integration by parts and obtain

If Re(|5)>0, [-yV^Cx+iy^xPl^O and thus we have PC(y;a,p) = ypC(y;a,p+l)+(l-a)pC(y;a-l,p+l). Interpreting this using co, we get (3)

co(y;a,p) = coCyja.p+^+Cl-^y^coCysa-l.p+l).

Iterating this formula n times, we have (4)

co(y;a,p) = £ £ = ( ) (£)(l-cc)(2-a)

(k-a)y-kco(y;a-k,p+n)

Now we choose a positive integer n so that Re(a-l) 0 for all (cc,P) e T, we have |co(y;a,P)| < S I T(p) |

= ILo

Lo ©r(k+Re(P))y-k.

From this the assertion of the lemma is clear. When there exists (a,p) e T such that Re(p) < 0, we choose a sufficiently large integer n so that Re(P+n) > 0 for all (a,P) e T. Then by (4), the proof in this case is reduced to the case already treated.

9.3. Functional equation of Eisenstein series

297

From the computation of the Fourier expansion we have done, we can at least determine the form of the Fourier expansion of Ek at various cusps. That is, we see that

where E' N (z,s;(a,b)) = y s I( m ,nKa,b) modN, (m,n)^o.O)(mz+n)"k | mz+n | "2s for (a,b)e (Z/NZ) 2 . We also see easily that for y e SL2(Z) E'k,N(z,s;(a,b)) I kY = E'k,N(z,s;(a,b)Y). Thus the Fourier expansion of E'k,N(z,s;(a,b)) at the Cusp Y(°°) is given by the Fourier expansion of E'k,N(z,s;(a,b)Y)- Thus we only need to compute E'k,N(z,s;(a,b)) for general (a,b). First writing E' k , N (z ) s;(a,b)) =

^

m

o

d

N> m

and then applying the Poisson summation formula to S(—TJ—;k+s,s), we have E'k,N(z,s;(a,b)) = c(s)ys+d(s)y1-k-s+ ^=1alVN(s)e(nz/N)co(47cny/N;k+s,s)

where c(s) and d(s) are meromorphic functions of s and an/N and bn/N are entire functions of s. Moreover, as long as s stays in a compact subset I an/N(s) I ^ A'n A and | bn/N(s) I < B'nB for suitable positive real numbers A, A\ B, B1. This shows Lemma 3. For any given compact subset T in R and y e SL2(Z), there exist positive numbers A and B such that if %*id Ek(z,s,%) I kY I ^ A(l+y" B ) as y -» °° as long as x = Re(z) G T 1-k and if % = id, the above estimate holds if T c R-{ -j-}. Exercise 3. Compute explicitly the Fourier expansion at ©o of E'k,N(z,s;(a,b)). We now make the following definition. For a C°°-function f: #"-» C satisfying flkY = f for all Y e T for a congruence subgroup T of SL2(Z), (5a) f is called slowly increasing if for any a e SL2(Z), there exist positive numbers A and B such that |flkOc(z)| °°; (5b) f is called rapidly decreasing if for any B G R and a e SL2(Z), there exists a positive constant A such that |flkOc(z)| °°.

298

9: Adelic Eisenstein series and Rankin products

Thus if f is slowly increasing, then for every a e SL2(Z), flkCX has polynomial growth in y as y - > oo. If f|k(x(z) for every a e SL2(Z) decreases exponentially with respect to y as y —» «>, f is rapidly decreasing. If g and f are both of weight k for F and f (resp. g) is rapidly decreasing (resp. slowly increasing), then M} of the cusp i°<> is isomorphic to an open disk with variable q (q = exp(27uz)) and since such neighborhoods are transformed by an element a e SL2(Z) to neighborhoods of a given cusp, | O(z) | is a bounded function on Y = PsH. Since Y has finite volume under y dxdy, the inner product (f,g) is well defined. Let us record this fact: (6)

Iff and g are C°°-class modular forms of weight k with respect to F and if f is rapidly decreasing and g is slowly increasing, then the integral defining the Petersson inner product (f,g) is absolutely convergent.

§9.4. Analytic continuation of Rankin products We start by introducing a slightly more general space of modular forms. Let % be a finite order Hecke character modulo N and %' be a character of (Z/NZ)X. We define Mk(N;%,%') to be the space of functions f satisfying (M2-3) in §1 and the following replacement of (Ml): for u € S(N)Ceo+ and a e GL2(F), f(axu) = Xf(u)xN(u)f(x)jk(ueo,i)"1,

(M'l)

((*

where a

% N (u)

is as in ( M l ) in §1 and

%'N

bY\

i

J l = %'((a- 1 d) N )

for

h\

J <= S(N)Coo+. If f G Mk(N;%,%') satisfies the cuspidal condition (S) in C

Qy

§1, f is called a cusp form and we write Sk(N;%,%') for the subspace consisting of cusp forms in Mk(N;%,%'). For each arithmetic Hecke character 0 : A X /Q X U(M)R + -» C x of finite order and f e Mk(N;x,%'), we define a new function f0 : GL 2 (A) ~> C by f®9(x) = 9(det(x))f(x). Then it is plain that f<8)0 G Mk([M,N];ze2,%feM"1), where [M,N] is the least common multiple of N and M and 0 M is the restriction of 0 to ZMX = n p |MZp X . Note that any character %' can be lifted uniquely to a finite order Hecke character 0 of AX/QXU(N)R+ so that %' = 0N- Then the association f h-» f®0 induces a map 0 0 : Mk(N;%,%') -» Mk(N;%02) whose inverse is given by 00" 1 . Thus we have

9.4. Analytic continuation of Rankin products

299

®0 : M k (N;x,0 N ) s M k (N;%0 2 ).

(1)

One can of course define the space M ^ ^ x ' ) for GL(2) over a totally real field for a character %': (O/m)x —» C x . However in this case, %' may not be liftable to an arithmetic Hecke character 0, and even if the lifting is possible, 0 may not be uniquely determined. Although we can construct a theory similar to the one presented here in the general case (see [H8]), this difficulty certainly adds more technicality in the treatment. This is the one of the reasons why we assumed that F = Q here. Now we look into the Fourier expansion of f e Mk(N;%,%!). We take the finite order Hecke character 0 such that 0N = %'. Then f<8>0 has the following Fourier expansion: f

®0((o

l

Thus applying 00" 1 , we find for y e Ax+ = AfxR+ that

Here we have used the fact that 0((£y)f) = 0(§y) = 0(y) = 0(yf) for y e and \ e Q+. Thus let us put (2)

A\

a(y;f) = ^(vfMyZjf®©) and ao(y;f) = e ^

Then the function of ideles y i-> a(y;f) no longer factors through ideals but really depends on the finite part of the idele y and satisfies (3a)

a(uy;f) = 0N-1(uN)a(y;f<8)0) for u e U(l) = Z x .

This shows the restriction of a(*;f) to ideles in (Af(N))x=(xe A f x | x p = l

if p l N }

factors through the ideals prime to N (thus f e Mk(N;%,%!) with non-trivial Nf is a GL(2)-analog of Hecke characters with non-trivial conductor). As for the constant term, we have (3b)

a o (auy;f) = Z!"1(u)a(y;f) for u e U(l) = Z x and a e Q + .

300

9: Adelic Eisenstein series and Rankin products

Thus the function y h-> ao(y;f) factors through AfX/U(N) = C1(N) instead of CL(1) for f®0. Now there is another operation. Let 0 be a primitive Dirichlet character of (Z/CZ) X (2 < C e Z). Then choose a representative set R in Zc x = n p | c Z p x modulo CZ C and define for f e Mk(N;%,x'),

(4a)

f|e(x) = C1G(e)XreRe"1(r)f(xa(r/C)) for a(r/C) = (J ' ^ j .

Then for u = I Q J E S([C,N]), ua(r/C) = a(ab"VC)u and thus f te(xu) = C^GCO) Xe^Wfixuair/Q)

= C^GCG) X

reR

reR

reR

Note that S([C2,N]) is generated by the u's as above, <x(v) for v e Z c (Zc = IIplcZp) and w e S(l) with w = l m o d C 2 . Since wa(r/C) = a(r/C)w' with w1 G S(N) for the w's as above, we know that f | 6 e M k ([C 2 ,N];%,%'0).

(4b)

We compute the Fourier expansion of f 10:

^fleffj fll =ao(y;f)L0"1(r)+ S ^ y . £vr Here note that by the definition of e (8.2.4), e(^=r) = exp(-27n[£yr/C]) for the fractional part fcyr/C] of ^yr/C in Q c x = n p | c Q P x . Hence 1

E

^

R

^

1

= 6(-^y)G(e- ) = e(^y)C" G(0) (see (4.2.6a)) if £ y Z is prime to CZ and otherwise it vanishes. This shows a(y;f | 0) = G(yc)a(y;f) and ao(y;f|0) = O. Thus we have (4c)

where we consider 0 is supported on ZQX extended 0 outside Zc x . Let % be a finite order character of Ax/U(N)Qoo+ = (Z/NZ)X. We write %* for the associated Dirichlet character modulo N. When confusion is unlikely, we write % for %*. We pick a Z[%]-algebra homomorphism

9.4. Analytic continuation of Rankin products

301

X : h k (To(N),x;Z[x]) -» C and consider its normalized eigenform f = iT_ X(T(n))q" e 5k(ro(N),%*). We may write the corresponding cusp form f e Sk(N,x) as f

((o i)) = X ^ Q + a ( ^ y z ; f ) e ( ^ x ) e ( i ^ )

for a(nZ;f) = X(T(n)).

Similarly we take another normalized eigenform g e M/(J,\|/) associated with a Z[\|/]-algebra homomorphism (p : h/(ro(J),\}/;Z[\|/]) —> C. Then we define the L-functionof X®(p<8>CG for an arithmetic Hecke character co (of Q) as in (7.4.2): (5)

L(s,A,®q>®G>) = L(2s+2-k-/,%\|/CG2) ]T ~=1 ^(T(n))(p(T(n))co*((n))n"s

= L(2s+2-k-/,% V co 2 )Xr =1 a(n;f)a(n;g)co*((n))n-s, which is an Euler product of degree 4 (see the proof of Lemma 5.4.2). Now taking the product *F(x) = (f(x)j k (x eo ,0)((g I G))(x)j/(Xoo,0) as a function on GL2(A), we consider the integral

(6)

Z(s,f,g,co) = J AX+/Q J A / Q T |Jj *)jd^(x)co(y) | y | Asd|ix(y),

where d|J. and d|Lix are the additive and multiplicative Haar measures discussed in §8.5. Disregarding the convergence of the integral, it is formally well defined. We will look into the convergence later and for the moment concentrate on computing it formally in terms of Dirichlet series. Let C be the conductor of CO and write O for the characteristic function of ZQX in Qc x . Then we see that a(y;f)ca(y;g I co)co(y) = O(yc)a(y;f)ca(y;g)co(y(c)), where a(y;f)° = a(y;f) and y ^ = yyc"1 for y e A x + . Then by the orthogonality relation (8.4.4):

we have, for Ax+ = Af x xR + (R + = {x s R | x > 0}) and Q+ = QflR+, Z(s,f,g,co) =

302

9: Adelic Eisenstein series and Rankin products = co(y))

JAK/O

= JA x <&(yc)a(y;f)ca(y;g)co(y) I y I A s e(2iy^)d^ x (y) * (yc)a(yf;f)ca(yf;g)co(yf) I y f I Asd|a.fx(yf)JR+e(2 iy»)yJd|i x (y). Thus we compute each integral, for U = U(l) &(yc)a(yf;f)ca(yf;g)co(yf) I yf | Asdnfx(yf)

<

= £ co * (n) A,c(T(n))9(T(n))n-s = L(2s+2-k-/,%" VcoV£(s,kc<S)(p<S>co) n=l

and

JR+e(2fyeo)yoosd^x(y) = J ^ e ^ y - M y = (4ic)-r(s). This shows (7)

L(2s+2-k-/,%-Vco2)Z(s,f,g,co) = (47t)-T(s)L(s,A,c®cpCG).

As already seen in §5.4, the series Z°° co*(n)A,c(T(n))(p(T(n))n~s actually conn=l

k+/ k verges absolutely either if Re(s) > I+-9- o r if Re(s) > / + j according as g is a cusp form or g is not a cusp form. The integration over A/Q needs no justification because A/Q is compact and the series are uniformly convergent. As for the integration over A x /Q x , replacing each term in the summation by its absolute value, we can perform the interchange of summation and integration because we are dealing with series with positive terms. The resulting series with positive terms after the interchange of the integrals converges if Re(s) is sufficiently large. Then we apply the dominated convergence theorem of Lebesgue quoted in §2.4 to perform the same interchange without taking absolute value. Thus the identity (7) is valid if Re(s) is sufficiently large. Now we compute the integral (6) differently. Recall that ¥ ( x ) = (f(x)j k (x M ,0)((g|co)(x)j i (x-,0). We consider a measure dv = I y I A'1d|ix(y)®d)J.(x) on

B(A)+ = U*Q jj e B(A) I y~ > Ok

9.4. Analytic continuation of Rankin products (y

Since L

x\

(y

J a ( u ) = a(yu)l

Q

iC\

II

303 r

A and d|i(yx) = | y | Ad^i(y) (because JxZpdUp

= I x | p (8.5.1)), b'*d|iB(b) = d|iB(b'b) = d|iB(b), i.e. d|iB is left invariant. Then we see, for B(Q)+ = B(A)+HB(Q), that Z(s,f,g,co) = JB(Q)+NB(A)+^(b)co(det(b))Ti(b)s+1d^B, where ri(x) is as in (1.9...). Let GL 2 (A) + = | X G GL2(A) | det(Xoo) > 0} and recall that C«>+ = {x e GL 2 (R) I det(x) > 0 and x(0 = /}. Now we choose a suitable Haar measure JUL on GL2(A)+ depending on SQL) as follows. Let S be any open compact subgroup of GL2(Af). Let djioo be the Haar measure on the compact group Coo+ZZoo (for the center Z«o of Goo+) with volume 1 (note that C00+/Z00 = T because C00+ = S02(R)Zoo). Then we define a measure |Lis on B(A)+SCoo+ such that

=J

B (A)

for all functions cp on B(A)+UCoo+, where d|Uo is the tensor product measure on SCoo+/Zoo of the Haar measure on the compact group S with volume 1 and the measure d|ioo on C00+/Z00. We take the measure on GL2(Q)+\GL2(A).f/SCoo+ induced by this measure. In fact, by taking a fundamental domain 7 of GL2(Q)+ in B(A)+SCoo+/SCo^, we define

This is possible because of GL2(A)+ = GL2(Q)+.B(A)+-S-Coo+ (Lib). The measure we have constructed depends on the choice of S in the following sense. If one takes an open compact subgroup S1 of S, then for any right S-invariant function f,

= [S:S']J G L 2 ( Q ) A G ( A ) + / S C o e + f(x)d^ s (x), because d|io f° r S1 is d|J,o for S multiplied by the index [S;S']. Let E(S) = Q x nSCoo + (which is either {±1} or {1}) and (p be a function on B(Q)+\GL2+(A)/SCoo+ such that cp is supported by B(A)+SCoo+ = B(Af)SGoo+. Then we have

We know from (1) and (4b) that, for % = x

X^CD"2,

F(xu)co(det(xu)) = ^#(u)xF(x)co(det(x)) for u e S(L)

304

9: Adelic Eisenstein series and Rankin products

where L = [C2,N,J] is the least common multiple of C2, N and J. Thus ^#(x)xF(x)co(det(x)) is left B(Q)+-invariant and right S(L)Co+-invariant. Note that ^ # (b) v F(b)co(det(b)) = xF(b)co(det(b)) for b e B(A) + ) and write Y = GL 2 (Q) + \GL 2 (A) + /SOo + . Applying (8a) to S = SQL) and E(S)={±1}, we have, writing djis as d|i.L and £, = XXJ/W 2 ,

We now compute T(TX) = (f(Yx)jk(Yx-O f(x)c((g | co-1)(x))jk(yx00,Ocj/(YXoo,0 = f(x)c((g | Then we see from (1.11) and (1.10a) that for w = (8b)

(y 0

L(2s+2-k-/,%-Vco2)Z(s,f,g,co) = L(2s+2-k-/,x'1\j/co2)JYfc(g I co-1)(w)co(det(w))

I co-1)(w)co(det(w))L(2s4-2-k-/,x-Vco2)E*(w,s-k+l,^) | y | A k d^ L (w) I co-1)(w)co(det(w))Ek./,L(w,s-k+l,O | y | Akd^iL(w). Here we claim that (9)

Y = GL2(Q)+\GL2(A)+/S(L)Coo+ = T 0 (L)W via

Xeo

h-> Xoo(/).

In fact, if Xoo = yx'ooU with u e Coo+, then ye S(L)GOo+nGL2(Q) = T 0 (L). Therefore the above map is well defined and surjective. The injectivity follows from the fact that GL 2(A)+= GL2(Q)+S(L)Goo+ (Lib). Thus the fundamental domain J we have taken in order to define djiL can be thought of as a fundamental domain of Fo(L)V# Then our measure on Y is the familiar one, y"2dxdy, by the definition of d|J.B. Noting the fact r\

~

, ] = y«>» we know

that (10) L(2s+2-k-/,%-Vco2)Z(s,f,g,co) = 2"x((g I co)EIk./,L(s-k+l,%\)/-1co-2),f)ro(L), where ( , )r is the Petersson inner product defined in §5.3 and g | co G f^(ro(J),\}/co2) (resp. f e 5k(Fo(N),%)) is the modular (resp. cusp) form corresponding to (g | co)<8>co (resp. f). By Lemma 3.3, E(z,s) =

9.4. Analytic continuation of Rankin products

305

E'k-/,L(s-k+l,%\i/"1co"2) has only polynomial growth as y - > s e P ! (Q) (i-e. E(z,s) is slowly increasing). The same fact is true for g | co (see Section 5.3). Since f is rapidly decreasing (see (5.3.8a)), f (g | co)E(z,s) is also rapidly decreasing, and hence ((g I co)Ef]C./fL(s),f) converges absolutely for any s. Thus the function assigning ((g | co)E'k_/,L(s),f) to s is a meromorphic function of s well defined for all s. In particular, when k * I, it is an entire function. This shows the analytic continuability of L(2s+2-k-/,%~1\|/a>2)Z(s,f,g,CQ). Thus we have Theorem 1 (residue formula). Let X : hk(Fo(N),%;Z[%]) -» C and (p : h/(ro(J),\|/;Z[\|/]) -> C be algebra homomorphisms with k > / and let co be a primitive Dirichlet character. Then L(s,A,c®(p®co) can be continued to a merok+/ morphic function on the whole complex s-plane and is entire if Re(s) > —— and if either %\(d2 Nk~l is non-trivial or X(T(p)) * co(p)(p(T(p)) for at least one prime p outside NJC. If X = (p and if % is primitive modulo N, the function L(s,Xc<8)X) has a simple pole at s = k whose residue is given by Ress=kL(s,?ic<sa) = 22k-17ik+1(k-l)!-1N-1np|N(l-p"1)(f,f)r0(N> where f e 5k(Fo(N),%) whose Fourier expansion is given by

Proof. The first assertion for non-trivial x V ^ 2 ^ " ' follows from Theorem 2.1 and the argument given above. Note that Ress=iGo,N(x,s,id) = Ress=iE0,N(x,s,id) = 7cn p |N(l-p" 1 )From this, we get, with a non-zero constant c, Ress=kL(s,^c<S>(p<8)co) = c(g I co,f)ro(L). We know from (5.3.10) that T(p)* = %(p)'1T(p), although we only proved this fact when N and J are powers of a prime in §5 (see [M, §4.5] for the general case). Thus co(p)(p(T(p))(g | co,f) = (g | co | T(p),f) = (g | co,f | T(p)*) = %(p)Xc(T(p))(g | co,f). Since ?ic(T(p)) = x(p)"^(T(p)) (see (5.4.1) and [M, (4.6.17)]), if co(p)cp(T(p)) ^ >.(T(p)), (g | co,f) = 0. When X = cp, we can easily compute c using the above residue formula of Eisenstein series and conclude with the residue formula in the theorem. Returning to the integral expression (8b), the integrand

306

9: Adelic Eisenstein series and Rankin products I co-1)(w)co(det(w))Ek./,L(w,s-k+l^) | y | A k

is the restriction of Lye{±i}B(Q)+^L2(Q)+xJ/(7x)co(det(7x))^#(7x)ri(7x)s+1 to B(A), which is left invariant under GL2(Q) and right invariant under S(L)Coo+. Note that the same fact is true for l°(g I co-1)(x)co(det(x)) | det(x) | A k E k _,, L (x,s-k+U). This shows that (11)

L(2s+2-k-/,x-Vco2)Z(s,f,g,co) co"1)(x)co(det(x)) | det(x) |

§9.5. Functional equations for Rankin products In this section, we prove the functional equation for Rankin products which also establishes the holomorphy of the L-function L(s,A,c®(p) if X * (p. We follow the treatment given in [H5, 1.9]. We start with algebra homomorphisms X : hk(r0(N),%;Z[%]) -» C and cp : h/(r o (J),\|/;Z[\|/]) -» C and suppose: (la) (lb) (lc)

k > /, % and \|/ are primitive modulo N and J, respectively, Z ' V is primitive modulo L,

where L is the least common multiple of N and J. The first assumption (la) is harmless, but the other two conditions impose a real restriction. To remove this assumption, the simplest way is the use of harmonic analysis on GL2(A) [J], although one can do that in classical way adding a large amount of technicality. The reason for this difficulty is that, without the conditions (lb,c), the L-function L(s,^c<8>cp) lacks some of the Euler factors (whose exact form can be predicted using Galois representations attached to modular forms discussed in §7.5; see [Dl, (1.2.1)]) at places p dividing L, and thus we cannot expect a good functional equation without supplementing missing Euler factors. In [J, IV], all the Euler factors are defined in terms of admissible representations and are computed explicitly when the attached local representations are subquotients of an induced representation of a character of a Borel subgroup. Then the functional equation is proven for any automorphic L-functions of GL(2)xGL(2) in [J,§19] including L(s,A,c<8>(p) treated here. The Euler factor for super-cuspidal local representations is recently computed in [GJ] (see also [Sch] and [H6]). Here we do not intend to be selfcontained. In fact, we shall use the semi-simplicity of hk(Fo(N),%;Q(%)) (for example, [M, Th.4.6.13]), when % is primitive modulo N, which is proven in the text as Theorem 5.3.2 when N is a prime power. Anyway, the proof in the

9.5. Functional equation of Rankin products

307

general case is basically the same as in the case of p-power level and is a good exercise after studying the proof in the special case. Let f (resp. g) be the normalized eigenform corresponding to X and (p. We write f and g for the corresponding classical modular forms fi and gi, respectively. We start with the integral expression 2-Tc(s)L(s,A,c<8>(p) = J Y (f°g)(x)E k _ a (x,s-k+l£) I det(x) | AkdHL(x), where Y = GL2(Q)+\GL2(A)+/S(L)Coo+ = T 0 (L)W and % = j ^ " 1 . Applying (y ^ the functional equation for Eisenstein series (Theorem 3.1): for w =

0

rc(s-/+l)Ek.a(w,s-k+U) (-l)G(^ 1 )rc(k-s)^(det(w))E k .,(wx L ,/-s^- 1 ), we have (2a)

2-T c (s-/+l)r c (s)L(s,X c ®(p)

, ^ | det(x) )r c (k-s) 1

xJ Y f g(xee)^(det(x))Ek./(xTL,/-s,^ ) | det(x) |

In order to avoid confusion between f(x)c = f (x) (complex conjugation applied to the value f(x)) and f°(x) (complex conjugation applied to the Fourier coefficient, i.e., a(n,f°) = a(n,f)c), we write the latter action as fc. Then we claim that f I V ( x ) = %(det(x))-1f(xxN-1) = N-k/2W(?t)fc(x) and g I xf 1 = r //2 W((p)g c for the constants xN =

W(A,) and W((p)

with absolute value

1, where

G GL2(Af). In fact, by the same computation as in (1.1a), we

see that (f I XN"1)! = f(XN(z))j(XN,z)'k regarding XN on the right-hand side as an element in GL2(Q). Then Proposition 5.5.1 shows the fact when N is a p-power. A key point is that T*(p) = XNT(P)XN"X for p outside N and XNS(N)XN -1 = S(N). The general case follows from the semi-simplicity of the Hecke algebra h k (Fo(N),%;Q(%)) by the same argument which proves

308

9: Adelic Eisenstein series and Rankin products (M(X)

0s]

Proposition 5.5.1. Note that TL = TNP with (3= A

for M(A,)N = L.

1

1

Then f | xL (x) = x t d e t W y ^ x f r V ) ) = f I TN^xp" ) = N"k/2fc(xp-1) using the fact that 1 = %(N) = %(Nf)x<~(N) = %(Nf). This shows that a(n;f | TL"1) = N-k/2W(X)a(n/M(X),f)c = N-k/2W(X)X(T(n/M(X)))c and a(n;g I TL 1 ) = J-//2W((p)a(n/M((p),g)c = J'//2W((p)9(T(n/M((p)))c fM(cp) 0\ for M((p) = L/J. Thus (2a) is equal to, for p1 = (2b)

(y and w = I

x\ I,

W(^) c W(9)2- 2 s + k + / - 1 L- 3 s + 2 k - 2 + 2 / N^ 2 r / / 2 ^(-l)G(^ 1 )r c (k^ xJ Y f c (wp- 1 ) c g c (wp- 1 )E' k . a (z,/-s,^ 1 )y k - 2 dxdy c

s

k/2

//2

= W(?i) W(9)2" N- J- L-

3s+2k 2+2

-

(z = x+iy) 1

^ oo (-l)G(^ )r c (k-s

1

xLL(k+/-2s,%V" ) XI=i MT(n/M(X)))9c(T(n/M((p)))ns+1-k"/, where we agree that X(T(n)) = cpc(T(n)) = 0 if n is not an integer. Since L is the least common multiple of N and J and since L = M(?i)N = M(cp)J, M(k) and M((p) are mutually prime. Then n/M(k) and n/M((p) are both integers if and only if n is divisible by M(X)M((p). Moreover M(k) \ J and M((p) | N, and hence we have T(pe) = T(p) e for p dividing the level (see (5.3.4a) and [M, Lemma 4.5.7]). Therefore, we have X(T(nM(q>))) = ^(T(n))X(T(M(q>))) and cp(T(nM(X))) = \(T(n))X(T(M(X))). Thus we see that (2c) X ^ i MT(n/M(?i)))(pc(T(n/M((p)))n-s n=1

?i(T(nM((p)))(pc(T(nM(?i)))n-s

X ^ 1 MT(n))(pc(T(n))ns. Combining (2a,b,c), we get 2-T c (s-/+l)r c (s)L(s,?t c ®cp) (^ x(M(X)M(9))s+1-k-/rc(k-s)rc(k+M-s)L(k+M-s,>,<8)9c). Using the fact that £(-1) = (-l)k"7, W(XC)W(X) = (-l) k (Proposition 5.5.1), M(X) = L/N and M(9) = L/J and (Exercise 2.3.5), we get

9.5. Functional equation of Rankin products

309

Theorem 1 (Functional equation). Suppose (la,b,c). Then for the least common multiple L of N and J, we have Goo(s)L(s,A,c®q>) = W O ^ X L N J ) - 8 * ^ ^ where

Goo(s) = r c (s-/+l)r c (s) and

By our assumptions (lb,c), we know that M(^) | J and M((p) | N. Under this circumstance, it is known that | W(A,c®q>) | = 1 (see [M, Th.4.6.17]). Since we know that Goo(s)L(s,^c®(p) is holomorphic if X * q> and

Re(s-k+l) > -f

<=> R e ( s ) >

^T

by Theorem 2.1. Since k > / > 0, this is enough to show that Goo(s)L(s,A,c®(p) is entire on the whole s-plane if X*
When X - (p, by the same reasoning,

the only singularity of rc(s-k+l)rc(s)L(s,^ c ®A,) is at s = k and k - 1 . Thus we have

Corollary 1. Suppose (la,b,c). Then Goo(s)L(s,^c®(p) is an entire function of s unless X = (p. If X = (p, Goo(s)L(s,^c®X) has two simple poles at s = k and k-1.

Chapter 10. Three variable p-adic Rankin products In this chapter, we first prove Shimura's algebraicity theorem for Rankin product L-functions L(s,A,<8>(p). Then we construct three variable p-adic Rankin products extending the result obtained in §7.4. As for the algebraicity theorem, we follow the treatment in [Sh3], [Sh4] and also [H5, §6], [H7]. We only treat the case of GL2(Q). For further study of this type of algebraicity questions for the algebraic group GL(2) over general fields, we refer to Shimura's papers [Sh5, Sh8, Shll, Shl2] and [H8] for totally real fields and [H9] for fields containing CM fields. As for the p-adic L-functions, we generalize the method developed in [H7]. Another method of dealing with this problem can be found in [H5]. The general case of GL(2) over totally real fields is treated in [H8]. There is one more method of getting p-adic continuation of L(s,X,®(p) along the cyclotomic line (i.e. varying s) found by Panchishkin [Pa].

§10.1. Differential operators of Maass and Shimura We study here the differential operators 8k introduced by MaaB and later studied by Shimura acting on C°°-class modular forms f on 0<\ for a complex number k and for y = Im(z) §k =

2 T C / ^ +2iy')

and

5 k = 8k+2r 2

"

§

k+25k, 8£f = f.

We also define (lb)

e = -T—y2— 2KI dz

and d = - — —. 2KI dz

It is easy to check that (lc) 8kf = y"kd(ykf) and 8Tk = y- k - 2r (y 2 dy- 2 ) r y k+2 . Now define a new "weight k" action (denoted by f i-> f||kOO of a e Goo+ as follows: f ||koc(z) = det(a) k/2 f(a(z))j(a,z)' k = det(a) 1 " (k/2) f | k a(z). Then the operators 8k and e have the following automorphic property: if f is of Cr-class, (2) 8 £ ( f | k a ) = (8 r k f)|| k+2 ra and e r (f|| k a) = (e r f)|| k -2 r a for all a e G^ + . The proof is a simple computation. We only give a detail account for 8. We may assume that det(oc) = 1 and need to prove 8k(f||kOO = (8kf)|k+2^« Note that ~ a ( z ) = j ( a , z ) " z , and ) ' k A and y(a(z)) = y(z)|j(a,z)|" 2 .

10.1. Differential operators of MaaB and Shimura

311

We see, writing j = j(a,z) and noting y(a(z)) =-^r that jj 2irf{8 k (£| k aH8 r k f)| k + 2 a} = C(z))..kr, iy J I 2iy

(2iyc , cz+d\l _ \ c z + d + cz+d Jj -

0 U

«

Exercise 1. Prove the second formula of (2). We now claim that the following two formulas holds: « •

Since the proof for these two formulas is basically the same, we only give an argument for the second formula. We proceed by induction on r. When r = l , the right-hand side equals

which shows the formula when r = 1. Now assume that r > 0 and that the formula is true for d r. Then

Note that d = 8 k + 2 j + ^ 1 and d(4jcy)j-r = (r-j)(4jcy)j - r - 1 . Thus 4jty

= t j=0

(n^j^{(r-j)(4Jtyy-r-18kf+(47tyy-r5k+1f+(k+2j)(4Jcyy-r-14f}. A

0+k)

The coefficient of (47cy)j " (r+1) 4f

is

S i v ^ n by, when 1 < j < r,

j+k)

-1)} _ /r+l\

=

Vj ) When j = 0, it is given by /r\F(k+r) /r\r(k+r) r

W r(k)

+r

/r+l\r(k+r+l)

^ r(k) " ^ °

Similarly, when j = r+1, it is given by r(k+r) _ ^r+l^r(k+r+l) T(k+r) ~ This shows the validity of (3).

O

312

10: Three variable p-adic Rankin products

Exercise 2. Give a detailed proof of the first formula of (3). If f is a holomorphic function on 9{ having q-expansion of the form f (z) = ^T-\ a ( n /N;f)q n (q = e(z/N)) for a positive integer N, then as a power series of q, f(q) converges absolutely on a small disk D r = { | q | < r } . Taking a smaller disk D e , f(q)/q gives a continuous function of q on the closure of D e which is compact. Hence | f(q)/q I is bounded, i.e. I f(z) I < Cexp(-27ty/N) as y —» «> for a constant C. In other words, f decreases exponentially at /°o. If f is a holomorphic cusp form for a congruence subgroup F of SL2(Z), the function f is therefore rapidly decreasing. Moreover 3rf — is also exponentially decreasing as y —> °°. Since (8£.f) is a polynomial in oz (4jcy)"1 with holomorphic function coefficients as above, (8£.f) still decreases exponentially as y —> «\ Since 5^(f ||kCx) = (8Tkf)||k+2rOc, applying the above argument to f||k(X for a e SL2(Z) in place of f, we know that f||k(x decreases exponentially as y —» °°. Thus we know that (4a) (S^f) is rapidly decreasing as a modular form for Y of weight k+2r if f is a holomorphic cusp form of weight k for Y. Similarly we can prove that (4b) (5£f) is slowly increasing as a modular form for Y of weight k+2r if f is a holomorphic modular form of weight k for Y. Let f = Zj=0(47cy)"Jfj be a C°°-function on 9l Suppose that the functions fj are all holomorphic. We claim that (5a) fj = O for all j if f = 0. We prove this by induction on r (see [ShlO, §2]). When r = 0, there is nothing to prove. Applying —, we get 0 = Z^J =1-(27CjV-T)(47iy)"^1fj. Then dividing dz out by (47iy)"2 and applying the induction assumption, we get fj = 0 for j > 0. This shows, at the same time, 0 = f = fo. In particular, Iterating this operation r times, we arrive, if f is of degree r in (47cy)4, at (5b)

8rf = crfr

with cr = (167i2)-rr!.

In particular, if f is an arbitrary C°°-function on 0{ with e r+1 f = 0, then fr = Cr'Vf is a holomorphic function, and er(f-(47cy)"rfr) = 0 by (5b). Thus by induction on r, we see the following fact for a C°°-function f on Prop.2.4]):

10.1. Differential operators of MaaB and Shimura e r+1 f = 0 <=> there exist holomorphic functions such that ^ J

(5c)

313

fj fj = 1, 2, ..., r)

where fj is uniquely determined by f. Now we suppose that f is a modular form of weight k+2r for a congruence subgroup T satisfying the equivalent conditions ii

fa

(5c). Then f||k+2rY = f for Y = r x

j=0

b^

J

G

r

r I"'*y/

i "~

•

implies

r \\

**yJ

i/||k+2ri

—

x

j=0

\

**j)

iViv /J\

*->*y*)

\\*£JT\X\

j=0

j=0

=

X(47iy)-Jfj(Y(z))(cz+d)-k-2r+J{c(z-2/y)+d}J

j=0

m=0

= (47ty)-rfr||kY+ S ( 4 i c y ) m=0

X j=m

Noting that the second sum in the last formula is a polynomial in (47cy)"1 of degree less than r, we conclude from (5a) that f r ||kY =r r for all ye T comparing the r(k) coefficient in (4jiy)"r. Now by (3), —-—77(8^) is a C°°-class modular form r(k)1 of degree r whose coefficient of a polynomial (47iy)" in weight (47iy)"r k+2r is fr. which Thus isf-CS^hj.) for hin r = ——— fr is of degree r-1. Repeating 1 (r + k)

this process, we can write, if k > 1 and r > 0, (6)

f = X r = o 8 ^2r-2j h J

if f

satisfies (5c) ([Sh3], [ShlO, Prop.3.4]),

where hj is a holomorphic modular form of weight k+2r-2j. The modular forms hj are uniquely determined by f. In this argument proving (6), we implicitly F(k) assumed that k > 1; otherwise, hr = —-—— fr may not be well defined (i.e. 1 (r + k) F(k) ——— may have a pole at non-positive integers k). Now we have by (2) I (r + k) er+1(f||k+2ra) = (e r + 1 f)| k . 2 a = 0 for as GL 2 (Q)nGoo + Then by (5c), we can write f ||k+2rOc = Zj=08k+2r-2jnj for holomorphic modular forms hj. On the other hand, by (2) J=0

U=0

jhJ l | k + 2 r a = J j=0

314

10: Three variable p-adic Rankin products

By the uniqueness of h'j, we see that hj = hj||k+2r-2j0c. This shows that (7)

f is slowly increasing (resp. rapidly decreasing) if and only if hj is holomorphic at every cusp (resp. a cusp form).

Let fA^k (To(N),%) be the space of C°° modular forms f satisfying: (Nl) (N2)

f is slowly increasing; e r+1 f = 0;

(N3)

flkY = Z(d)f for y = (* J) e r o (N),

where % is a Dirichlet character modulo N. We also consider the subspace !A££(ro(N),%) of 5\£k(ro(N),x) consisting of rapidly decreasing forms. Modular forms in these spaces are called nearly holomorphic modular forms. We have proven Theorem 1. Suppose that r > 0 and k > 1. Then we have

The isomorphisms are given by fh-» (hj) as in (6). isomorphisms are equivariant under the " ||" -action of Goo+.

Moreover these

We write the projection map f H» ho as (8a)

H :

^

+

which is called the holomorphic projection. Since we have a Galois action f H> f° ( a e Aut(C)) taking fWk(r0(N),x) to a4(r o (N),x°) (see §§5.3 and 5.4), we can define a Galois action on fAtk+2r(ro(N),x) so that f° corresponds (hja) under the isomorphism of Theorem 1. Then by definition, we in particular have, if k > 1 and r > 0 , (8b)

H(f°) = (H(f)) a for all f e ^ + 2 r ( r 0 ( N ) , x ) .

Note here that

d(E°° a n q n ) = S°° n=0

n a n q n , i.e.

d = q—.

n=0

This implies

Q(J

d(f°) = (df) a for the naive Galois action on q-expansion coefficients. By our definition, the monomial "(4rcy)~m" is invariant under the Galois action. Since 5k is the sum of the multiplication by -k(47iy)"1 and d, we see that (8c)

8 k (f°) = (8 k f)° for f e ^ + 2 r ( r 0 ( N ) , x ) .

10.1. Differential operators of MaaB and Shimura

315

We want to compute the adjoint of 8 k and 8 under the Peters son inner product. We start with a general argument. Consider a compactly supported C°°-function <>| and aC°°-function \\f. Let T7

o

2 3

^

k

, 3 .

_

2 3

T-

• 2 ^

E = 2 y x — , A = 2{^?+-T-}, E x = y z —, E y = ly 37, dz ^y 3z 3x ^ oy A

and

f

Ax = —, Ay = - / { - + — } . J dx y ay

Then E = E x +E y and A = Ax+Ay. Writing \|/c(z) = (\|/(z))c for complex conjugation c, we have, for y(z) = Im(z),

^

ay This shows

^Vy+k
dy

dy

0 0 = -/ Thus we have the following adjoint formula: (9) Let F be a congruence subgroup of SL2(Z). Let <>| be a C°°-class modular form of weight k. We take a sufficiently fine (locally finite) open covering FVtf= UjeiUj on 0< so that Uj is simply connected. Let n : ^ - > I\H be the projection. We choose a connected component Uj* in TE'^UJ). Thus n induces an isomorphism: Uj* = Uj. Then we take a partition of unity of class C°°: 1 = EJGIXJ; here, %j is a C°°-function with Supp(%j) c Uj. We regard %j as a function on U*j under the identification induced by TC. Then %j(|): Uj* -> C is a C°°-function whose support is contained in Uj*. Thus we can extend Xj4> t 0 all iH by 0 outside Uj*. Then for any C°°-class modular form \\f of weight k+2 for F, we see from (9) that

(10)

J

r w J G l

X

/

j k

2

= (4>,e\|0

316

10: Three variable p-adic Rankin products

as long as £jeiJ#Sk(%j<|>)Vcykdxdy is absolutely convergent. When either <>| or \j/ is rapidly decreasing and the other is slowly increasing, we can always choose the covering {Uj} so that the above sum is absolutely convergent. Let us see that this is so. We take out a small neighborhood of each cusp in T\H and write the rest as Yo. Then Yo is relatively compact. We choose a closed neighborhood D s of each cusp s so that Y o r i D s * 0 , r w = LJDSUYO and i : DO = S^O^oo) (for sufficiently large M > 0 ) inducing i*(y"2dxdy) = t~2dtd0 for the variable 8 on S 1 and t on [M,oo). Here i is given by an element a of SL 2 (Z) such that cc(s) = /«> (then t = Im(a(z)) and G = Re(cc(z))). Now take a covering S 1 =ViUV2 so that Vi is isomorphic to an open interval. Then U y = USfiij = ViX(M+j-8,M+j+8) for (1/2) < 8 < 1 covers D s . If F(t,0) decreases exponentially as t —> <*>, we see easily that Zijluj jFt"2dtd0 converges absolutely. We then choose a finite open covering Yo = UaU a - Then the covering {Us>ij, U a } does the job. Thus we have Theorem 2. Suppose either f e ^ ( r o ( N ) , z ) or g e ^k+2( r o(N),%) is rapidly decreasing. Then we have (8kf,g) = (f,£g)Corollary 1. Suppose that f e 5 k (r 0 (N),%) and g e r < k/2, then (f,g) = (f,H(g)).

fl£k(ro(N),x).

If

Proof. Writing g = H(g)+^ =1 8^_ 2j hj, we see that ( f ^ . ^ j ) = (ef, S ^ j h j ) = 0 since ef = 0. This implies (f,g) = (f,H(g)). Now we see that 27i/8k+s( , (CZ+d))," 2s) = ^ - ( ccz + d ) - k I cz+d | -2s-(k+s)c(cz+d)"k-11 cz+d | "2s I cz+d 1 (z-z )

k 2 - 1 cz+d I -2^\ = •^ ± |
(11)

Bk+

Thus iterating the formula (12a) we have

8k+s((cz+d)-k | cz+d I "2s) = (-47cy)-1(k+s)(cz+d)-k-21 cz+d I "2(s-1),

10.2. The algebraicity theorem for Rankin products

317

(12b) 8^ + s ((cz+d)" k | cz+d | " 2s) = (-47ty)- r r ( S + k + r ) (cz+d)- k - 2 r I cz+d | "2(s"r). r(s+k) Recalling the definition in §9.3, E'k,N(z,S,X) = y

S

S

(

m

1

k

'

'

^

we know that (13)

^ 5 ^ E ' k , L ( z , s , % ) ) = E' k+ 2r,L(z,s-r,%). (-47c) r -^ T(s+k+r)

§10.2. The algebraicity theorem for Rankin products We now recall (9.4.7) and (9.4.10): (1)

(4rc)-T(s)L(s,r®q>) = 2- 1 (gE 1 k . a (s-k+l,xV 1 ).f)r 0 (L),

where we have used the notation in §9.4, which we recall briefly: X : h k (r 0 (N),%;Z[%]) -> C (resp. q> : h/(r o (J),v;Z[v]) -> C) i s a Z f t ] algebra (resp. Z[\\f]-algebra) homomorphism; f and g are associated normalized eigenforms, f = E~ A,(T(n))qn and g = I^_1(p(T(n))q11; L is the least common multiple of N and J. We consider the set (P consisting of all Z[%']-algebra homomorphisms X* : h k (ro(N'),%';Z[%']) -> C varying N1 and %' but fixing k. For X* e (P, the integer N1 is called the level of X\ An element X : hk(Fo(N),%;Z[x]) -> C e (P is called primitive if N is minimal among the levels of the homomorphisms in the following set: {^'e 2>U'(T(p)) = X(T(p)) for all but finitely many primes p}. The primitive element X as above is uniquely determined by X1 and is called a primitive homomorphism associated to X\ The level of the primitive element A, is called the conductor of X\ We note that 5t'(T(n)) = A,(T(n)) if n is prime to the level of X\ The modular form f associated to a primitive X is called a primitive form. This notion of primitive forms coincides with the one given in [M,p.l64], and we refer to [M, 4.6.12-14] for the proof of the above facts. We now assume that X is primitive. If the reader is not familiar with the notion of primitive forms, he may make the stronger assumption that % is primitive of conductor N; in this case, f is automatically primitive in the above sense [M, §4.6]. Under the assumption of primitivity, we have (2)

fc||kTN = W(?ic)f for W(kc)e

Cx with W(XC)W(X) = %(-l),

318

10: Three variable p-adic Rankin products

(o -n where TN = N 0 • This assertion is proven in Proposition 5.5.1 when % is primitive modulo p r and N = p r for a prime p. A proof of the general case can be found in [M, Th.4.6.15]. Hereafter we simply write (f,h)L for (f,h)ro(L)Then we have, for any smooth modular form h on Fo(L) with character %* which is slowly increasing at every cusp of Fo(L), (3a) This follows from the fact shown below (5.3.9): (h||kTL,f) = (h,fIlk^L1) and XL1 = -tL- In particular, W(X)W(kf(f,f)N = (f lk*N,f |k*N)N = ( f ^N and thus (3b) W(X)W(?i) c = 1. We write

T L / N : ^ k ( r 0 ( L ) , % * ) -» ^ k ( r o ( N ) , % * ) for the adjoint of fL/N 0^ [Fo(N)pro(L)] with p = I I. We use the same symbol TL/N to denote

the corresponding operator Mk(L,%) -> Mk(N,%). Thus (3c) We see that

(h | T L/N ,f) N = (h,f I [r o (N)pr o (L)]) L . a

b"|p .

j

= fa

[

Lb/N^i andhence

d JJ

ro(N)

This implies that r o (N)pr o (L) = r o (N)p and h | [r o (N)pr o (L)](z) = (L/N)k"1h(Lz/N). Then using these formulas, we have 2(4JC)-T(S)L(S,?L C ®(P) = ( g E ' t a C s - k + L z V 1 ) . ^

xi/-1) ||k-/xL),f ||kxL)L = (L/N)1-(k/2\(g||/xL)(E'k./,L(s-k+l,xx|/-1)lk-ftL),(f lkt N ) I tr o (N)pr o (L)]) L |T L /N,f||kX N )N

x({(gc(Lz/J)(E'k./,L(s-k+l,xr1)||k-rXL)}lTL/N,fc)N, where we have used the following formula: I, I, ,. f L / J CA „, g||/XL = g||/Xj||^ Q J = (L/J) //2 W(9) gc (Lz/J). As seen in §§9.1 and 9.2 ((9.2.4a,b)), we can compute the Fourier expansion of ) and Gk-/,L(0>£) explicitly. To recall this, we put

10.2. The algebraicity theorem for Rankin products

319

where the norm character N only has symbolic meaning but later we consider N to be the cyclotomic character of Gal(Q/Q). Then we have, for

C = zk T(k)

and a Dirichlet character ^ modulo L (4a) G k , L (0£) = C{8k,12-1LL(0^)-8k,28^id(8TC I y I A )- 1 cp(L)L- 1 +G L (^ k )}, and for C = ik2kLkA% (4b) G k , L (l-k£) = C{8k,28Ltl(87C | y | A where 8 k j is the Kronecker symbol and S^id = 1 or 0 according as t, = id or not. Now we write h* = h : GL 2 (A) —> C for the adelic modular form corresponding to h (see (Ml) in §9.1), i.e., h(au) = %L(u)h(Uoo(0)Jk(Uc«,0"1 for u e S(L)Goo+ and a e GL 2 (Q). As seen in §9.3 (noting h|| k tL = L 1-(k/2) h | kXL), we have (5a) (E'u(z and by (1.13), (5b)

E ! k . a (z,s-r,% ¥ - 1 ) = (-

Solving the equations c

= x(-l)W(k ) (6a)

s-r = m-k+1 and s = l-k+/+2r, we conclude from that for an integer m with / < m = / + r < ^ p 1

)||k./TL)} |TL/N,f||kxN)N

1

)|k+/.2mt0 L)

Now we note that

k+/ We see from (4b) for an integer / < m = /+r < -y- that

(6b)

r(m+1 -l)T(m)L(mX® (?) = (L/N)1"(k/2)2"1(47c)mr(m+l-/) x({(g||/XL)(E!k-/,L(m-k+l,x\|/-1)|k_/TL)} |TL/N,f ||kxN)N = \|/(- i)L m + 1 - k N ( k / 2 H J-//2(-47c)m-/(47c)mW(9)W(X,c) x({g(Lz/J)(8^//_2mGk+/-2m,L(z,l-k-/+2m,%"V))} = t({g(Lz/J)(8^ / / _ 2 m E' L (rV^ k + / " 2 m ))}

320

10: Three variable p-adic Rankin products t = /k+/2k-/+2m7C2m-/+1L/-mN(k/2)-1J"//2W((p)W(?ic)

where

and E ' L ( x V ^ k + / " 2 m ) = 8k+/.2m,25L,i(87i I y I A )- 1 +E L (x'ViV k + / " 2 m ). Now we treat the other half of the critical values L(m,A,c<E>(p) for m with k+/ l+-"2" < m < k. We solve the equations s-r = m-k+1 and s = 0 and get m = k-l-r with r > 0. We have (7a)

2(47t)-mr(m)L(m,?tc(p) = (L/N) 1 - (k/2) ({(g||/T L )(E' k . a (m-k+l,xV' 1 )||k-nL)}

2.k4X^^Xr*)

II2m+2-k-/tL) )

where we have used the following formula:

Then we see from W(X)C = %(-l)W(Xc) that (7b)

r(m+1 -l

=

2 k-/ + 2m / k + / 7C 2m + l-/ L /-m N (k/2)-l J -//2 w((p)w()l c )

-1))} |T L/N ,f c ) N , where G ' L ( ^ k ) = 6k,i2-1LL(0^)-5k,25^id(87i I y I ^ " Lemma 1. Let % be a Dirichlet character modulo N. Suppose that all the prime factors of L divide N. Then for all h e 44(r o (L),%) and a e Aut(C), we have T L/N (h) a = T L/N (h a ). Proof. Since T = T L/N is the adjoint of [r o (N)|3ro(L)] for (3 = 1

Q

J,

we see for two modular forms (|) of weight k for T = Fo(L) and \\f of weight k for T = T0(N) that

Jrv^V I kPyk"2dxdy = J ^ | T¥yk'2dxdy. Using this formula, we compute T directly: I kPyk"2dxdy = (L/N)k-1Jrx^)Xi/(p(z))yk-2dxdy = (L/N)-1Jprp.1N^)(pl(z))\|/(z)yk-2dxdy

10.2.

The algebraicity theorem for Rankin products

321

I kPlY(z)V(z)yk~2dxdy. Thus we see that 0 IT = I y e prp.lxr,(t) | k(3ly. Note that p r

prp^nr = {(* *] e ro(N) I b e

(L/NJZJ

=

Thus choosing a common representative set R for

(p l )- 1 rp l nr\r t = prp-1nriNTi we have r o (L)pT o (N) = lJYERro(L)ply and r o (N)pT o (N) = This shows that the operator T = T L /N coincides with T(L/N) on and, if all the prime divisors of L divide N, then T is equal to T(L/N) on the larger space f^4(ro(L),%). In fact, if all prime divisors of L divide N, for any fa b^ Y= , in F , taking 0 < u < L/N such that au = -b mod (L/N), we see

that Y( 0

x

I e p r p ^ n r . Thus we can take R =\ I Q

x

11 0 < u < L / m as

the common representative set, and we see TL/N = T(L/N): a(n,f | T) = a(nL/N,f). This shows the lemma if all the prime divisors of L divide N. The assertion of the above lemma is in fact true without assuming any condition. However, to prove it, we need a fairly long argument either from the theory of primitive forms [M, §4.6] (see [Sh3] for the actual argument) or from the cohomology theory studied in §7.2. We shall prove it in the general case by a cohomological argument. First we introduce the space of N-old forms. Since

p t ^ r o W p t ^ r o C N t ) for p t = ( j J j w i t h 0 < t e Z , for any modular form f e ^k(To(N),%), we see that ft(z) = f(tz) = t1-kf | p t is an element of ^k(ro(tN),%). Then we define the sub space S^(T o(L),%) of N-old forms in 5k(ro(L)>%) to be a subspace spanned by modular forms in the set: {ft(z) = f ( t z ) | 0 < t | L/N and f e 5 k (r o (N),%)}. Lemma 2. Let % be a Dirichlet character modulo N. Then for all h E 5kT(r0(L),%) and a e Aut(C), we have TL/N(h)° = T L/N (h a ). Proof. What we need to prove is TL/N(ft)a = TL/N(fta) for any divisor t of L/N and all f e 5k(Fo(N),%). Writing the operator fh->f| k p t as [t], we see that [L/N] = II p [p e ] according to the prime decomposition L/N = EL | L/NP6* ^ n e n b v definition of TL/N» we have TL/N = n p | L / N Tpe, where Tpe is the adjoint of [pe] : 5k(Fo(L/pe),%) —> 5k(ro(L),%) which has the same effect as, for example, the adjoint of

322

10: Three variable p-adic Rankin products

[pe] : A(r o (N),x) -> 5 k (r 0 (Np e ),x). Thus we may assume that L/N is a prime power without losing generality. Then Tpe = Tp,e0Tp,e-i0-*-°Tp,i, where Tpo- is the adjoint of [p] : ^ O W p J - 1 ) , * ) -> 5 k (r 0 (NpJ),z). Thus we may assume that e = 1 because fpk | Tpo- = fpk-i if j > 0 and k > 0. If p IN, the assertion is already proven by Lemma 1. Thus we may assume that p^N. Then choosing x e SL2(Z) by the strong approximation theorem (Lemma

(o -n 6.1.1) so that x = 1 1

0

I modp and x e Fo(N), we can take the following

set as the representative set R as in the proof of Lemma 1:

i] I o < u <

Then

IQ

Jx = I

I = x(5p modp

shows that f I T p = f I T N (p)

if

f G 5k(ro(N),%). On the other hand, we see that

OVl n This shows that fp | T p = (l+pk"1)f. This finishes the proof. The following fact is a key to our argument: Proposition 1. Suppose that k > 2. Then the orthogonal projection

K% : 5k(r0(L),%) -> 5^(ro(L),x) w rational; that is, we have 7Txa(f°) = {7Cx(f)}° for all a e Aut(C). Proof. Note that (3tXL = tXL/t for any divisor t of L. This shows that ^(%) = ^(TQ(L),X) is stable under XL. Note that the Hecke operators T(n) of level N and L are different on 5k(Fo(N),x) if n has a prime factor q such that q divides L but q is prime to N. To indicate this difference, we write TiXn) for Hecke operators of level L if necessary. We write L = NoLo such that Lo is prime to N and all the prime factors of No divides N. If n and tLo are relatively prime, we see a(m,ft | T L (n)) =

£ % L ( b ^ a ^ f ) = a(m,(f | TN(n))t) 0
b

l

« f t | T L ( n ) = (f|T N (n)) t where the subscript "L" to % is given to indicate that x is a Dirichlet character modulo L (i.e. XL(^0 = 0 if b and L are not relatively prime). This follows from the two facts

11 mn/b 2 <=> 11 m and x L ( b ) = ° <=*

10.2. The algebraicity theorem for Rankin products

323

under the assumption that n and N are relatively prime. When t is a prime power q r |Lo with r > 1, then a(m,fqr I T L (q)) = a(mq,fqr) = aOm/q^f) = a(m,fqr-i). If r = 0, a(m,f | T L (q)) = a(mq,f) = a(m,f | TN(q))-%N(q)qk-1a(rn,fq). These facts show that S(%) is stable under TL(n) for all n > 0. Since 5(%) is stable under XL and TL is an automorphism of S(%), 5(%) is stable under TLTLWTL" 1 = Ti/n)* (= the adjoint of Ti/n) under the Petersson inner product). Let SH%) be. the orthogonal complement of 5(%) in 5k(Fo(L),%) under the Petersson inner product. The stability of S(%) under TL(n)* and TL(n) for all n shows the stability of 5X(%) under these operators. Then, for any Q(%)-subalgebra A of C, we let /^(A) (resp. /t(A)) denote the A-algebra generated over A in EndcCS-Kx)) (resp. Endc(5(%))) by TL(n) for all n. By the duality between 5k(F0(L),x;A) and h(A) = hk(r0(L),%;A) (Theorem 6.3.2), we know that h(C) = /t-L(C)®/t(C) as an algebra direct sum. Now we can think of corresponding subspaces H5X(A) and H5(A) in Hp(Fo(L),L(n,%;A)). Thus |3t induces a morphism [pd = [r o (N)p t r o (L)] : H^roOTJ^njfcA)) -* Hi>(ro(L),L(njc;A)), and H5(A) is defined to be the sum of Im([pt]) over all positive divisors of L/N. Then from the fact that H1P(r0(N),L(n,x;Q(%)))(8)Q(x)C = H1P(r0(N),L(n>x;C)), we know that H5(Q(%))®Q( X )C = H5(C). We can define H5X(A) to be the orthogonal complement of H5(A) under the pairing (6.2.3a) in the full cohomology group Hp(r o (L),L(n,x;A)). The formula (6.2.3b) then tells us that the EichlerShimura isomorphism induces isomorphisms of Hecke modules, UsHC) = SHx)®SHjf)c and H5(C) = 5(X)05(XC)C, where, for example, SH%°)° = { f (z) | f € 51(%c)} for complex conjugation c. This implies (see the proof of Theorem 6.3.2) 6HC) = AX(Q(X))®Q(%)C and /t(C) = ACQ(X))<8>Q(Z)C.

Thus again by the duality theorem, we know that SHr,Q(X))®Q(x)C = SHX), 5(X;Q(X))®Q(Z)C = S(%) and 5k(r o (L),x;Q(x)) = 5x(x;Q(X))e5(%;Q(x)), where SH%;A) = 5 k (r 0 (L),x;A)ri5 1 (x) and 5(x;A) = 5 k (r 0 (L),x;A)n5(x). This shows the rationality of nx over Q(x)- Now we want to show that for a e Gal(Q(x)/Q), the following diagram is commutative:

324

10: Three variable p-adic Rankin products KX : 5 k (r 0 (L),x;Q(%)) -» 5k(r o (L),x;Q(x))

(*)

la

4 a

a

KXO : 5 k (r 0 (L),x ;Q(X)) To see this, we consider the Galois action of a E Gal(Q(%)/Q) on the ro(N)-module L(n,%;Q(%)), which induces an isomorphism a : H 1P(ro(N),L(n,x;Q(%)))=H 1p(r o(N),L(n,x a;Q(%))), which in turn induces a Q-algebra isomorphism

a* : hk(ro(N)oca;Q(x)) = hk(r0(N) JC; Here a* takes T(n) to T(n) = a"1T(n)a and coincides with a"1 on Q(%). It is easy to check that (h a *,f) a = (h,f a ) for h e h k ( r 0 ( N ) , x ° ; Q ( z ) ) and f e 5 k (r 0 (N),%;Q(x)), because a* takes T(n) to T(n). Then again by the duality, we know the commutativity of (*), which shows the proposition. Corollary

1. For

a G Aut(C), we have

fe

^ k +2r( r o(L),%)

(k > 1 and r > 0)

and

a

(f | T L / N ) = f° IT L/N .

Proof. Using Theorem 1.1, we can write

f

= E r ^

Then, with the notation of the proof of Lemma 1, we see that

= i ; = o I Y e R (L/N)2J(8i+2r_2j(hj|TL/N)). Note that TL/N is the adjoint of [L/N]: 5k(ro(N),%) -> 5k(ro(L),%) and thus Ker(TL/N) contains Ker(jcz) in Proposition 1. Thus TL/N = T L / N 0 ^ - Then by Lemma 2 and Proposition 1, we see that (hj I T L / N ) a = (%(hj) I T L/N )° = (7tx(hja) | TUN) = hj a IT L/N . Then by (1.8c), we know that

(f|TL/N)°= { ^ = 0 I Y 6

Now we shall prove the following algebraicity theorem for Rankin products:

10.2. The algebraicity theorem for Rankin products

325

Theorem 1 (Shimura [Sh3,4]). Let X : h k (r 0 (N),%;Z[%]) -> C (resp. cp : h/(ro(J),\|/;Z[\|/]) -» C) be a Z[%]-algebra (resp. Z[y]-algebra) homomorphism and let f = E°°_ ^(T(n))q n e 5k(ro(N),%). Suppose that X and (p are both primitive. Then for all integers m with I <m
and for all ae Gal(Q/Q), S(m,?ic(p)a = S(m,A,ac Q(XC) has a section in the category of Q-algebras; i.e., there exists an algebra homomorphism s : Q(^ c ) -» hk(ro(N),%c;Q(A,c)) such that X°s = id and B = Ker(X) is a Q-subalgebra of hk(r0(N),%c;Q(X,c)). Thus h k (r o (N),x c ;Q(^ c )) = Q(A,)Uc0B with an idempotent l^c When % is primitive (modulo N), hk(ro(N),%c;Q(^c)) is semi-simple and thus this is obvious. The general case is shown in [M, Th.4.6.12]. We consider the linear form l\ : <|>i-* (Uc,(|)> on 5 k (r 0 (N),% c ;Q). The action of a on L(n,%c;Q) induces an isomorphism a : HJ>(ro(N),L(n,%c;Q)) -> H ^ r o C N J . L C n ^ Q ) ) . We see that aT(n)a -1 = T(n) and thus induces an algebra isomorphism a : Then we see by definition that (ha,<|)a) = (h,<|))0. In particular, we have a

) , i.e., hc(<\>)° = lXco(
As a linear form, l%c is the unique one satisfying /^c(<)) | T(n)) = X(T(n))c/xc((t)) and /xp(fc) = 1 because of the duality between the Hecke algebra and the space of modular forms (Theorem 6.3.2). We consider another linear form <(> h-» (<|>,fc). Since (<|> I T(n),f c ) = (<|),fc | T(n)*) = ( l\c is a constant multiple of <>| h-> (<|),fc), i.e. l\c(§) = C((|),fc). We compute C. We see that 1 = /xc(fc) = c(fc,fc) = C(f,f) and thus

Here the equality z h-> - z . Writing

(fc,fc) = (f>f)

can be shown via the change of variable

326

10: Three variable p-adic Rankin products i

C

•= 1

8

2

L

rr """' , .,; 2m m

lr /

UN l

)) I T L / N )

2

k+r

if / S m S -«-,

we know from the above formula that $ has algebraic Fourier coefficients and

Here we have used the fact fca = (f°)c shown after the proof of Corollary 5.4.3 when N is a prime power. The assertion in the general case also follows from the same argument by [M, Th.4.6.12 and (4.6.17)] under the primitivity assumption on X. Then by Lemma 1 and (1.8b,c), we have, for £ = %"V>

tfWm^^L^))^)

if l+^<

m < k,

" [H( a (L/J)(5^|E'(^A^ k+/2m )) | TL/N) if / < m This shows Ll-mWlS(m,Xc®) because for example pkA = X(T(p)2)-?t(T(p2)) (see (5.3.4a) and §6.3). Applying the above theorem to the primitive algebra homomorphism q>®co : h/(ro(Jl)>\|/c°2'Z[xl/co2]) —» C for a Dirichlet character co modulo C such that (pco(T(n)) = co(n)(p(T(n)) for all n prime to CJ, we have Corollary 2. Let the notation be as in the theorem. Then for each Dirichlet character co modulo C, write T for the level of the primitive algebra homomorphism q><8>CG associated to g | co. (T is a divisor of the least common multiple of J and C 2 .j Then we have, for all integers m with I < m < k, (l)m+/ _ and for all GGGal(Q/Q), S(m,A,c(pco)°: = S(m,X,oc®q>a®a)a). §10.3. Two variable A-adic Eisenstein series We fix a prime p and put p = 4 if p = 2 and p = p otherwise. Let O be the integer ring of a finite extension K of Q p . Let t, and % be two Dirichlet characters modulo p. We assume that %(-l) = 1. We agree to put in force the following convention:

10.3. Two variable A-adic Eisenstein series

327

^(n) = %(n) = 0 when p | n, even if % or £ is trivial. We also consider the continuous character x X given by K(Z) = (1+X) s(z) for s(z) K = K X : W = 1+pZp -> A = O[[X]] s given by u ^ = z (u = 1+p). We have added the suffix "X" to K because we want to write Kz for the "same" character having values in Az = O[[Z]] obtained from K by replacing X by Z. We consider the following formal q-expansions:

E(%iVk)(q) =

where i is the trivial character modulo p, Ep(%Nk) is the q-expansion introduced in §10.2 and (n) = nco(n)"1 e W. Thus the operation f 11 is just taking out terms involving q11 for n divisible by p, i.e. f | i = f-(f | T(p))(pz). This series has the following property: (El)

E(^%)(u k -l,e(u)u r -l) = dr(E(%co"kA^k)|e^co-r) for k > l and r > 0 ,

where d = qrz = — — and e is a finite order character of W and the above identity is the identity of power series in O[e][[q]]. Here we denote by O[e] the subring of Q p generated over O by the values of e. In fact, for each character e : W —> Q p x of finite order, regarding e as a character of Z p x by e(n) = e«n», we have K«n))(e(u)uk-1) =nkeco"k(n) and thus

= d r { IZ=i ^co- r (n)Xo Q p x of finite order, we consider the form E(^,x)(X,e(u)-l). Then by (El), we have E(^,x)(u k -l,e(u)-l) = E(%Kok) I ^e € f*4(r o (p a p),e^ 2 %) for sufficiently large a independent of k. Now we take a finite extension M of the quotient field L of A = O[[Y]] and write J for the integral closure of A = O[[Y]] in M. Take a J-adic form G e S(\|/,J). We define a convolution product

328

10: Three variable p-adic Rankin products

Write %P = e'%co"k and \|/Q = e'^co' 7 for Q E J4.(J) and P e ,#(A) when k / PflA = P k)£ ' and QflA = P/>e». Then, if 0 < r < -y-, we have ve l 2T ~ ) | ^co"r), and if ^ - < r < k-Z G*E($A|T1x)(P,Q,e(u)ur-l) = G(Q)dr(E(VQ-1%Pe-2co2riVk-/-2r) | ^eo)-r) = G(Q)d-r-1+k-/d2r+/-k+1(E(\|/Q-1xPe-2co2riVk-/-2r) = G(Q)dk-/-r-1(G(\|/Q-1xPe-2co2rN2r"k+/+ Then viewing G*E(£,\|/'1%)(e(u)-l) as a formal q-expansion with coefficients in A<§>oiJ for a fixed finite order character e : W -» Q p x , we know that G*E(^,\(/"1x)(e(u)-l) satisfies, if k>Z, G*E(§,\|T1x)(P>Q,e(u)-l) = G(Q)(E(TI) | §e) € ^ k (r 0 (p a )^ 2 x P ), where T] = V Q ^ Z p e ' 2 © 2 1 ^ " 7 and a = max(f(P),f(Q))+2f(e) for the conductors pf^P), p f ^ and pf^e^ of %P, \|/Q and e, respectively. Thus we know from (7.4.5) that

Here the suffix " a " indicates that we regard the character (£2%)a = £2% as defined modulo p a ; thus, the character (^2%)a may not be primitive. By the definition of completed tensor products, S(%eo;A)®oJ is the completion of U § under the m-adic topology for the maximal ideal m of A(§> ad. Lemma 1. Let S be the space of H = S°° a(n;H)(Z)qn with n=l

a(n;H)(Z)e (A®^)[[Z]] satisfying H(e(u)-1) e S(%oo;A)®oJ[£(u)] for all finite order characters e 6>/ W. Then, for H e S, we /zave H e S(%oo;A)®oJ[[Zl] and hence H(e(u)ur-1) G SCXoojA)®^- Moreover, we have H | e(e(u)ur-l) = H(e(u)ur-1) | e for all finite order characters e and r > 0. Proof. By the same argument as the one given below (7.4.5), we may assume that J = Zp[[Y]] and A = Zp[[X]]. Then A®oJ = Zp[[X,Y]]. Let £n be a primitive pn-th root of unity. We put n = ria(Z+l-£nO) e Z[Z], where ^ n ° runs over all conjugates of £n over Q. Then we know that con = (Z+l) pn -l = IIo<m
10.3. Two variable A-adic Eisenstein series

329

Thus H H» ^ n=0 H(Cn-l) defines a morphism cpn : S -> ^n=0S(Xoo;A)® J[Cn]- If (pn(H) = O, a(n;H) is divisible by (Z+l-£ m ) in Zp[Cm][[X,Y,Z]] for all m < n . Thus H e conS. This shows that Ker(cpn) = conS. Since HnCOnS = {0}, we know that S injects into Km Im((pn) = lim (S/conS). n

The space S(Xoo;A)<§>oJ[[Z]]/(cGn) consists of H n = £~ =1 a(n;H n )q n with a(n;H n )e Zp[[X,Y]][Z]/(con) such that H n (C m -l) e S(%~,A)®0j&m] for all 0 < m < n. Thus S/conS is a subspace of S(%00;A)®0J[[Z]]/(con). Thus we have a natural map of S into lim S(Xoo;A)(g)oJ[[Z]]/(cGn), which is equal to n

S(Xoo;A)%>aJ[[Z\l Since S(x~;A)<§)oJ[[Z]] is a subspace of Zp[[X,Y,Z]][[q]], the map is injective, and hence we can regard S as a subspace of S(Xoo;A)<§)oJ[[Z]]. Then the projector e : SOc^A) -> S ^ x ^ A ) extends linearly to e : S(Xoo;A)<§>oJ[[Z]] -*

(Proposition 7.3.1) S ^ x ^

By definition, e commutes with the specialization map: H h-> H | z =z for any z e Q p . In particular, H(e(u)ur-1) |e = H|e(e(u)u r -l) in Zp[e][[X,Y]][[q]]. Corollary 1. Suppose that G(Q) has q-expansion coefficients in Q and G(Q)e 5 k (r 0 (p p ),\l/Q;Q). Then the two limits lim>G(Q)dr(E(\|/Q-1Xp8-2co2riVk"/-2r) I £eco"r) I T(p) n! for 0 < r < ^ , Mm G(Q)dk-/-r-1(G(\i/Q-1xPe"2co2r^2r-k+/+2) I ^eco"r) I T(p)n! for

^ <

r < k-/,

exist in O[e][[q]] under the p-adic topology, where Moreover, writing the above limit as h, we have, in O[e][[q]], h = (G*E(^,xi/-1X) I e)(P,Q,e(u)u r -l),

which is an element in 5k(ro(p a p),Xp^ 2 ; Q)

for

a

such

that

Proof. We start with a general argument. Let A be a Q-subalgebra of C and ge

^(roCp^xyV^A).

Write

g = 2™0{ATzyY%

p

j

with

gj

e A[[q]].

Take f e jy£?' (r o (p ),\|/;A) and write f = I™0(4jcy)- fj. We consider

g( 5rk. f | e^) e *$?££ If k'+/ > 2m+2m', we can write uniquely

(ro(pp),x;A).

330

10: Three variable p-adic Rankin products

f I e£) = H(g(8£

I<

for hj G 5k-+/-2j(ro(pP),X;A) and H(g(8£f I e$)) e 5 k - + /(r o (p P ),z;A). Then equating the constant terms of (*) as a polynomial in (4?cy)'1, we see from (1.3) that (1)

godr'(fo I e§) = H(g(5rk.f I

This shows that H(g(8£ f I e£)) = godr'(fo I e ^ - Z ^ ' ^ h j . Now assume that A is a subalgebra of QflK. Then we can consider the identity (1) as an identity in Q P [[q]]. Since H(g(8k'f I e§)) e 5 k ' + /(r o (p P ),%^ 2 ;Q p ), we see, under the p-adic topology, that H(g(5rk. f | e$)) I e = Km {g(8£ (f | e£)) I T(p) n! }, where T(p) acts on q-expansions as in the corollary. ^ e P"YO[[q]] ( 0 < y e Z),

Note that, for

d I T(p) n ! = £ ~ = 1 a(mpn!;d^))qn = £ ~ = 1 mpn!a(mpn!;(t))qn -> 0 as n —>• eo. Since hj e p"YO[[q]] for sufficiently large y by Theorem 6.3.2, we conclude that

(2)

H(g(8£f I e^)) I e = {godr'(fo I e?)} I e.

By (2.4a,b), we find that E^N*) and G(^N^) are modular forms except when j = 2 and ^ = i d . The forms E(N2) and G(N2) show up in the formula when either k = /+2r+2 or k = 2r+/. In this case, m = 0 and m1 = 1 by (2.4a,b); thus, we can find f e ^ 2 ( r o ( p ) , Q ) such that f0 = E(A^2) or G(N 2 ). Hence the existence of the limit follows from the above argument. When k'+/ > 2m, H(f(8J g I e^)) | e is always a classical modular form. Thus to finish the proof, we only need to check the identity h = (G*E(£,r|) | e)(P,Q,e(u)ur-l) for T| = y\fAX- Taking f as above and writing g = G(Q), we recall (1): G*E(^\|r 1 x)(P,Q,e(u)u r -l) = gdr'(f0 I e§) = H(g(5rk,f

\e$))+^=T+T'd"hy

By Lemma 1, (G*E(^,\|/-1%)(Z) | z=e(u)ur-i) I e = (G*E(^,\j/-1%) | e)(Z) | z=e(u)ur-i. Thus (G*E(§f>|r1x) I e)(Z) I z=e(u)ur-i I e = ( G ^ E ^ , ^ ^ ) I e)(Z) I z=e(u)ur-i. Note that G*E£,\fh) I e € S ^ f o ^ A ) ® ^ ^ ] ] . For F G S((X^ 2 ) C O ;A), we have F I e = lim F | T(p)n! under the w-adic topology for the maximal ideal of A. In fact, assuming F E S((%^ 2 ) a ;A) for some a > 0, we have by definition,

10.4. Three variable p-adic Rankin products

331

F | e(u k -l) = f(u k -l) | e = l i m F ( u M ) I T(p) n! under the p-adic topology on 5k(ro(pa),%^2co"k) for k > a with a sufficiently large a. This implies, writing = I I " Pb, F | e = lim lim (F mod Pa>a) | T(p) n! , which is equal to the

pa

limit under the m-adic topology. Thus we see that l

X) I e = Km((G*E(§,y ^ I T(p)n!.

This implies

(G*E(§,\|r1x) I e)(P,Q,e(u)ur-l) = ,1m(G*Eft,>|r1z)(P,Q,e(u)ur-l)) I T(p)n!, which is equal to h. Here the last limit is taken in O[e][[q]] under the p-adic topology.

§10.4. Three variable p-adic Rankin products We take two normalized eigenforms G e S(\|/;J) and F G Sord(%;l). Note that Sord(X;l) = 0 for p less than 11 (see §7.6). Thus we may assume that p > 5. We define the A-algebra homomorphisms X: hord(x;A) —> I and (p: h(\|/;A) —» J by F | T(n) = A,(T(n))F and GI T(n) =
the ring O is integrally closed in I and J.

Then I <§> c>J[[Z]] is an integral domain finite and flat over O[[X,Y,Z]]. We write A = I <§> oJ[[Z]] and its field of fractions as M = Frac(A). Then we consider Sord(%;l)<§>oJ[[Z]] = Sord(%;l ®oJ[[Z]]). As constructed in §7.4, we have an inner product (, )| : S ord (%;l)xS ord (%;l)-> K for the quotient field K of I. We extend this product linearly to an inner product ( , )A : S ord (%;A)xS ord 0c;A) -> M. Now we define Lp(?t(pc) = (F,e(G*E(id,\i/"1x)))A e M. We study the values Lp(X<8Xpc)(P,Q,R) for (P,Q,R) e *(l)x*(J)x*(A). We write PflA = Pk,£., QflA = P/)£», R = Pr>E and Xp = e X C0 k» ¥p = e'V®"'* ^ e further write ^ P : h k (r 0 (N),x P ;Z[x P ]) -^ Q and 9 Q : h k (r 0 (p p ),\)/ Q ;Z[\|/ Q ]) -> Q for the algebra homomorphism given by F(P) | T(n) = XP(T(n))F(P) (resp. G(Q) I T(n) =
332

10: Three variable p-adic Rankin products

as long as Q is admissible relative to G. By the ordinarily of F, we have Xp is either trivial with N = p or primitive modulo N. To compute the value Lp(A,®(pc)(P,Q,R), we assume the following condition on e: (P) G(Q) I eco"r is a primitive form of exact level J for a p-power J. Although the condition (P) is hard to verify without using representation theory ([Ge] and [C]), let us explain a little bit about this assumption, referring to [H5, II, Lemma 5.2] for details. By the definition of primitive forms given at the beginning of §10.2, we can always find a primitive form g of level J° associated with G(Q) | eco"r. We know that g | T(n) = eco"r(n)?ip(T(n))g for all n prime to p. If p I J° and g | T(p) = 0, we see that g = G(Q) I eco'r because eco"r(p)^P(T(p)) = 0. For almost all non-trivial e, J° is divisible by p and g I T(p) = 0. Thus the assumption (P) is known to be true for almost all e. When G(Q) is ordinary of level J (and thus G is automatically ordinary), the condition P is equivalent to (P1)

eco"r is neither XJ/Q"1 nor i p if \|/Q is non-trivial, and eco"r * i p if \\fp is trivial, where i p is the trivial character modulo p.

We state our main result in this section: Theorem 1 (Three variable interpolation). Let X : h ord (%;l) -> I be an I -algebra homomorphism associated with a normalized eigenform F e Sord(%,l) and G be a normalized eigenform in S(\j/;J). For each admissible point Q of #(J), we write (pQ : hk(Q)(r0(pP),eQ\|/0)"k(Q);O[e]) -> Q p for the O[e]-algebra homomorphism associated with G(Q). Then we have a unique p-adic L-function Lp(X®(pc) in the quotient field of l<§>oJ[[Z]] with the following evaluation property: for P e A(l) with k = k(P), admissible Q e #(J) with I = k(Q) and R = Pr>e e A(A) satisfying the condition (P) and 0 < r < k-l, we have S(P)Lp(A,(pc)(P,Q,R)

(27cO k+/+2r ^ 1 ' k (F(P)°,F(P)°)

'

where F(P)° is the primitive form associated with F(P), No is the level of J is the level of the primitive form G(Q)|eco~r and 1 if either %p is non-trivial or k = 2,

S(P) = i

J ^ _ } - i ( 1 . _E -i MT(p))2 XP(T(p))2

otherwtse.

10.4. Three variable p-adic Rankin products

333

Here we understand W(k?) = 1 if %p is trivial and k > 2. The relation between this three variable p-adic L-function and those obtained already (Theorems 7.4.1 and 7.4.2) will be clarified in the following section. We can even compute the value Lp(^(8>(pc)(P,Q,R) without assuming (P). However the computation adds additional technical difficulty. Thus here we only present the result under (P). We refer to [H5, II, Th.5.1d] for details in the general case. We start proving the above theorem. We want to relate the value Lp(A,®(pc)(P,Q,R) with the complex L-value L(SOA P ®
and Case II: ^ y ^ < r < k-/.

By definition, writing A for Qp(^Pc®(pQ®e), we have by Corollary 3.1 (1) =

Lp(?t(pc)(P,Q,R) f (F(P),e(G(Q)dr(E(\|/Q-1%Pe-2co2rNk-/-2r) | eo)-r)))A in Case I, 1 (F(P),e(G(Q)dk-/-r-1(G(\|/Q"1xPe"2co2riV2r+/-k+2) | eo)-r)))A in Case II.

Now we prove a lemma: Lemma

1. Let

e : (Z/p^Z)* —> Ox

be a character.

Assume

that

lim (f(g I e)) I T(p)n! converges to a power series e(f(g | e)) Radically in 0[[q]] for

f

and

ge

O[[q]],

where

I ~ = 1 a n q n I T(p) = I ~ = 1 a p n q n

and

ST^anq111 e = IT = i e(n)a n q n . Then lirn^f I e)g) I T(p)n! converges to a power series e((f I e)g) and satisfies

e(f(g | e)) = e(-l)e((f | e)g).

Proof. We define I JT_ anqn | p = Supn | an | p . Then the p-adic convergence is just the convergence under the norm I | p in 0[[q]]. Writing a(n,f) for the coefficient in q11 of f, we first prove that a(np r ,f(g|e)) = e(-l)a(np r ,(f|e)g) if r > y . This is just a computation, a(np r ,f(g|e))-e(-l)a(np r ,(f|e)g)

= X Je(npr-j)a(j,f)a(npr-j,g) - e ( - l ) £ Jr£(j)aa,f)a(npr-j,g) = 0, because e(npr-j) = e(-j) = e(-l)e(j) if r > y. This shows (f(g | e)) | T(p)r = e(-l)((f | e)g) | T(p)r. This implies that ]JmJ(f \ e)g) I T(p)n! and Krnjfig I e)) I T(p)n! converge at the same time and e(f(g 18)) = e(-l)e((f I e)g).

334

10: Three variable p-adic Rankin products

Note the fact that e(G(Q)drg) = Km (G(Q)drg) | T(p) n! in O[[ql] under the padic topology. Then, writing simply E = E(£ A^"'"2*) and F = 8j,12-1LL(0^)-5j,25^id(87C I y I A )- 1 ( P (L)L- 1 +G(^ j ) for £ = \|/Q"^pe"2co2r and j = 2r+/-k+2, we know, from (1) and the fact that eco-r(-l) = (-l) r , Lpacpc)(P,Q,R) = (-l) r xj ( p ( p )' e ( G (Q) I eco- r d r E(^ k - / - 2 r ))) A

in Case I,

1 (F(P),e(G(Q) | eco- r d k - / - r - 1 G(^ 2r+/ " k+2 ))) A in Case II, J (F(P),e(H(G(Q) | eco-r5rk_/_2rE)))A X

r

r

in Case I,

!

[ (F(P),e(H(G(Q) | eco- 8|r-i7_ kl2E )))A in Case II,

where we have used (3.2) and the fact (2.4a,b) that E and E1 are classical modular forms to obtain the last equality. Since e is self-adjoint under the pairing ( , ) A and F(P) | e = F(P), we have , 1 M 1 (F(P),H(G(Q) I eco"r5rk_/_2rE))A in Case I, T n r, cw cD n m Lp(A,<8Xp )(P,Q,R) = (-1) xs , f . \ P ^ [(F(P),H(G(Q)|eo)-r5kr-i7_rki2E'))A in Case II. Write L for the least common multiple of N = p a (the level of F(P)) and j = pP+2Y (the level of G(Q) I eco"r), and put p 6 = L/N. Then the self-adjointness of T(p6) shows that (2)

Lp(?t<S>(pc)(P,Q,R)

8 [(F(P),H(G(Q)|eco- r 8l_/- 2r E)|T(p 5 )) A in Case I, = (-l)\(T(p))" x vi I * P (F(P),H(G(Q) eco- r 5^7_ r ki 2 E') T(p 8 )) A in Case II, 8

1 A (T( ))-5x{ ( F ^ ' H ( G ( Q > I «»"r8k-/-2rB) I T(p 8 ))c in Case I, P P [(F(P) ) H(G(Q)|eo)- r 5| r -i7_ r ki 2 E')|T(p 8 )) c in Case II. When either %p is primitive modulo N = p a or k = 2, as we have already seen in (7.4.1a) (and in a remark after Corollary 7.2.1 when k = 2) that (g,F(P)) (F(P),g)oo (g,F(P) c h) ( F ( P) jF(p ))oo - ( F ( p ) j F ( p ) c | T ) - (F(P),F(P))' because F(P)C | x = W(A,C)F(P). We now compute (F(P),g)c in terms of the complex Peters son product ( , ) when %p is trivial. The above formula is true except for the last term. In fact, the linear form (F(P))

10.4. Three variable p-adic Rankin products

335

satisfies L<>T(n) = XP(T(n))L and L(F(P)) = 1. Thus by the duality theorem (Theorem 6.3.2), we find that L(g) = (F(P),g)c. Therefore we need to compute

i

f° -^

F(P)C I x for x = Q . As shown in Corollary 7.2.1 (or its proof), if k > 2, there exists a unique normalized eigenform F(P)° e 5k(SL2(Z)) such that F(P)° | T(n) = XP(T(n))F(P)° for n prime to p and F(P)° | T(p) = (cc+p)F(P)° for a = A.p(T(p)) and P = j>k'lX?(J(p))'1. We note here that p c = a because a+P is a real number (because it is an eigenvalue of the self-adjoint operator T(p)). Moreover

F(P) = F(P)°-a-1F(P)°lk[o i } We call this form F(P)° the primitive form associated with F(P). Either when %p is primitive modulo p a or when k = 2, we simply put F(P)° = F(P), i.e., F(P) is already primitive. We put W(k) = 1 in this case, W(k) e C because F(P)° 11

1

Q

I = F(P)°. Then we see, noting that (F(P)°)C = F(P)°, that

F(P) c |x = F ( P ) c | ( °

^

= F(P)°lk[0 1J-ap-1F(P)o = -ap-1(F(P)o-a-1pF(P)°lk[0 J ) . Writing (, ) p for the Petersson inner product of level p and ( , ) for that of

fP Q\

level 1, we now compute, for 8 = I Q A and f = F(P)°, (F(P),F(P)C | x)p = -pp'^f-a^f I S.f-a^pf I k 8) p . Now we use the fact that (f,f)p = (SL2(Z):r0(p))(f,f) = (l+p)(f,f), (f,f| k 8) p = (f | T(p),f) = (a+p)(f,f), (f I k8,f)p = (a+p)(f,f) (see the proof of Lemma 2.1), (f Ik8,f I k 8) p = (f I kx,f | k x)p = P k2 (fl|kx,f|| k x) p = p k " 2 (l+p)(f,f), where Tp(p) in the last formula indicates the Hecke operator of level p. This shows, if x p is trivial and k > 2, that (3)

(F(P),F(P) C | x) p =

We now compute (F(P),H(G(Q) | eco^S^^^E) | T ( p 6 ) ) c in Case I. We suppose either k = 2 or %p is non-trivial. We recall (2.6a) replacing X and cp there by Xpc and (pQc: (4) 2(47c)-/-T(/+r)L(/+r,Xp®(pQc(8)8-1cor) = (L/N) 1 - (k/2) (-4Tc)T(r+l)- 1 % p (-l)W(?ip) X({(G(Q)C I e" 1 ^||/T L )(^. / . 2 r (E' k . / . 2 r , L (z,j,^ 1 ) ||k./-2rXL))} I T( P 5 ),F(P)) N

336

10: Three variable p-adic Rankin products

where j = l-k+/+2r, \ = Xp^Q^e^co21 and L is the least common multiple of N = p a and J = p^ +2y which is the level of G(Q) I eco"r. As already seen in the proof of Lemma 2.1, the operator TL/N is nothing but T(p ). We want to relate the value (F(P),H(G(Q) I eco-r6^.2rE) I T(p 5 )) c to that of (4). We have (5)

(F(P)°,F(P)°)(F(P),H(G(Q) I e c o " ^ . ^ ) I T(p 5 )) c = (H(G(Q) | e c o - ^ . ^ E ) | T(p6),F(P)) | T(ps),F(P)) l/XL^.^E} | T(p5),F(P)).

We need to compute G(Q) I eco"r || /XL. Under (P), we have (by (2.2)) (6) = (L/J)//2W(cpQ(8)eco-r)(G(Q)c I e' for a constant W((pQ®eco~r) with absolute value 1. Then (5) is equal to (-l)/(L/J)//2W(9Q®eco-r)({G(Q)c I e^co^z/J)} ||flL5}E(§ tfj) I T(p5),F(P)), where j = k-/-2r. Then recalling the formula (2.4b), (7)

(E' k . / . 2r , L (z,l-k+/+2r,^ 1 ) ||kXL)*

and the fact that

D(s,F(P),gc(pvz)) = ]T~=1 e W = Xp(T(p)) v p- vs D(s,F(P),g c ), we see from (4) that (5) is equal to

xr(r+l)r(/+r)L(/+r,?ip(S)(pQc<8)e"1cor). Thus finally we get, if either %p is non-trivial or k = 2 and if (P) holds, (8) Lp(?i(pc)(P,Q,R) = (-l) / Xp(T(N))^ P (T(J))" 1 J (//2)+r N 1 " (k/2) W(Xp)" 1 W((p Q ®eco- r ) r(r+l)r(/+r)L(/+rAp(g)(poc(g)£-1CQr) (27c/)k+/+2r7C1-k(F(P)°,F(P)°)

10.4. Three variable p-adic Rankin products

337

We will show that the above formula holds also in Case II if either %p is non-trivial or k = 2 and if (P) holds. Now we consider the case where %p is trivial and k > 2, but we still remain in Case I. We then need to compute (H(G(Q) I eco-r8rk./-2rE) I T(p 5 ),F(P) c I x) P

= p(k/2)-i(G(Q) | e c Q - ' I ^ L S ^ E ) | T(p5),F(P)c||kT)p Q

x({(G(Q)c |e^coOCLz/Dl/TL^.^} I T(p5),F(P)c||kx)p 1 k+/+2r

-

L- (k - 2r)/2+/+1 7C"V ( ^ 2) ' 1 r //2 W((p Q ®e^

x({(G(Q)c | B-lGf)£f)HxL^.^E'k./.^LCza-k+Mr,^1)||xL}

| T(p5),F(P)c||x)p.

By (2.6a), we have, for g = (G(Q)C | e"V)(L2yj) W-2rj^^

We then know, noting that N = p and k is even, that (H(G(Q) | e c o - ^ . ^ E ) | T(p 5 ),F(P) c I 2 and %p is trivial, under (P), that (9)

Lpa<8Xpc)(P,Q,R) = (Ay x(1 x(1

ykA \ P (T(p)) 2

yi(1 y ( 1

pk-2 \

2

\ P (T(p)) 2

(27r/)k+/+2r7C1-k(F(P)°,F(P)°)

In this case, for XP° : h k (SL 2 (Z);Z) -> Q given by F(P)°|T(n) = V(T(n))F(P)°, we note that W(?ip°) = 1. We will see later that (9) also holds without any change in Case II if %p is trivial and k > 2. We now deal with Case II. As before, we first assume that either %p is trivial or k = 2. We compute (F(P)°,F(P)°)(F(P),H(G(Q) | eco-^^.'k^E 1 ) I T(p 5 )) c = (H(G(Q) | e c o - ^ . ^ E ' ) | T(p5),F(P))N = (-l)/(L/J)//2W((pQ®eco-r) x({(G(Q)c I e-VXLz/J)|/r L G(Q) I e Q - ^ ^ E 1 } I T(p6),F(P))N.

338

10: Three variable p-adic Rankin products

Now by (2.7b), solving k-l-m = k-/-r-l and 2m+2-k-/ = 2r+/-k+2, we have m = /+r and ^(T(L/J))(L/J)-/-T(r+l)r(/+r)L(/+r,^p®(pQc<8>e-1cor) =

2 k + / + 2r / k + / JC / + 2 I+ l L -r-(//2) N (k^)-l wap)

x({((G(Q)c Ie-lG?)0.z/T)\\tih)(^l;£+2E')}

| TL/N,F(P)°)N.

Thus we have (F(P)°,F(P)°)(F(P),H(G(Q) | eco-r = (-l)'+r(27cO"k'/"2rJck"1N-(k/2)+1X.p(T(L/J))Jz/2+rW((pQ®eco-r)W(X,p)-1 1

which shows that the formula (8) again holds in Case II if %p is non-trivial or k = 2. We now assume that k > 2 and Xp is trivial. Thus N = p. We compute (H(G(Q) | eco-r6t!tk+2E') I T(P8),F(P)C I T) N = (H(G(Q) | eco-r8k;|tk+2E') I T(P8),F(P)C = C({(G(Q)C|e-^OCLz/Dl/XLS^-^E'} |T(p5),F(P)c||kT)N, for C = ( - ^ ' ( L / ^ ' V ^ ^ W C ^ e c o " 1 ) . By (2.7a), for g = (G(Q)C I e V 2(47c)-/-r(L/J)-/-rXp(T(L/J))r(r+l)r(/+r)L(/+rAp€>q)Qc<8)e-1O)r) = (L/N)1-(k/2)(-4jt)k-1-/"T(/+2r+2-k)

where we have used (2.4a) and (2.5a) combined with (E' k>L (z,s,^ 1 ) ||T L )* = (.l) k+/ r(/-k+2r+2)- 1 (27i0 / " k+2r+2 2L- (/ " k+2r+2)/2 E l .

This shows that (H(G(Q) | e c o ^ ^ E 1 ) I T(p5),F(P)c | x) N = (-l)/+r(27c/)'k"/'2r7ck-1J(//2)+rW(9Q(8)eco-r)Xp(T(L/J)) xr(r+l)r(/+r)L(/+r,A,p(8)(pQc®e"1cor). Thus again, in Case II, under (P), we know that (9) remains true if %p is trivial and k > 2. This finishes the proof.

10.5. Relation to two variable p-adic Rankin products

339

§10.5. Relation to two variable p-adic Rankin products In this section, we clarify the relation between three variable p-adic Rankin product and the two variable one constructed in §7.4. To relate two p-adic L-functions, we need to perform a supplementary computation to ease a little bit the condition (P) in §4. For that, we use the notation introduced in §4. We assume that G is ordinary and \|/Q is non-trivial. Let i p be the trivial character modulo p. Then G(Q) I t P = G(Q)-G(Q) I T(p)(pz) = G(Q)-(pQ(T(p))p1-/G(Q) 1,8 for 8 = I 0

j I. Then writing the exact level of G(Q) as Jo and assuming that

G(Q) is primitive and Jo is divisible by p, we see for J = Jop and x =

0 -1

that G(Q) Upjl/x = J1-(//2){G(Q)-(pQ(T(p))p1-/G(Q) = p(//2)W((pQ)G(Q)c(pz)-(pQ(T(p))p-//2W((pQ)G(Q)c. Performing the same computation as in §4 replacing the formula (4.6) by G(Q) I tpl/TL = (L/J) //2 {p 1 -(^W«p Q )G(Q) c I /8-p k- 2 J0 (k/2) - 1W((pQ )G(Q) c}(^), we have (1)

Lp(X(pc)(P,Q,R) = (-l)/Xp(T(N))?ip(T(Jo))-1Jo(//2)+rN1-(k/2)W(V"1W(cpQ) r(r+l)r(/+r)E(/+r)L(/+r,X,p(g)(poc) (27cOk+/+2r7t1"k(F(P)°,F(P)0) ' S-l

where

E(s) = 1--

\ 1

p

>Q(T(p)Ap(T(p))j Let us prove (1). We only deal with Case I because Case II can be done similarly. We have from (4.5) that (F(P)°,F(P)0)(F(P),H(G(Q) I ipSkj.&E) I T(p 8 )) c . ^ } Q

ITL/N,F(P))

Q

x([{G(Q)c(Lz/J)-(pQ(T(p))-1p'G(Q)c(Lz/Jo)} 1 ^ . , . ^ ] I T L /N,F(P)), which is equal to (by (4.4) and (4.7))

340

10: Three variable p-adic Rankin products

xr(/+r)r(r+l)L(/+r,A,p®(pQc) xL p" (Xp(T(L/J))(L/J)-/-r-(pQ(T(p))cp?lp(T(L/Jo))(L/Jo)-/-r) ==(-l)/(-l)r7Ck"1(27iO"k"/"2rW(9Q)W(^p)"1Jo//2+rN1"(k/2)>.p(^ x(l-(pQ(T(p))-c^p(T(p))-1p/+r-1)r(/+r)r(r+l)L(/+r,)ip®9Qc). /+r

/

Then by (4.2), we have the formula (1). We now assume (Ql) ^ V Q ^ X P is primitive modulo L, (Q2) %p is non-trivial and eco"r is trivial, (Q3) \|/Q is non-trivial. We write the level of the primitive form G(Q) as Jo, which is divisible by p by (Q3). We have defined in §7.4 the two variable Rankin product Lp(A.c®(p). We had the following evaluation formula (Theorem 7.4.2): Lp(Xc®(p)(P,Q) = L We recall the functional equation (Theorem 9.5.1): (27c)"2k+1+/r(k-/)r(k-l)L(k-l,>,pC(8)(pQ) =

Q

where w = (-l) / W(V)W(9 Q )G(%p-V Q )^p(T(L/Jo))L 1 -V 2 (p Q (T(L/N)) c N / - (k/2) . This shows that Lp(A.c9)(P,Q) = (-l)k+/W(V)W((pQ)^p(T(L/Jo))L/-1Jo//2(pQ(T(L/N))cN/-(k/2) (27tOK+'jc1"K(F(P)°)F(P)°) By (1), we know that Lp(^®(pc)(P,Q,P0)

= E(/)Lp(Xc®cp)(P,Q),

10.5. Relation to two variable p-adic Rankin products where

2 E(Z) = ( 1 ) = (1- ° F ). 9Q(T(p)Ap(T(p)) VT())

j^ j

and

341

Here we note that

E ( p Q ) = (1 _M3M)^

Thus

writing

^>(T(p)) Lp(?i<8>(pc)p0 for the element of the quotient field of I <§> QJ given by evaluating the variable R of Lp(^<8>(pc) at Po, we expect to have L p (k®(p c )p 0 = E.Lp(A,c(p) in F r a c ( l ® o J ) . To see this, we note the following fact valid in a more general setting. Let A be an integral domain and !P b e a set of prime ideals with the property that ripG2>P= {0}.

Then the congruence

a = (J m o d P

for all

P e fP ( a ,

p G A) implies the identity a = (3. If T is a set of infinitely many primes of height 1 in A, this % satisfies the above property flpepP = {0} because for Pi, ..., P r G (P, P i O • -flPr is contained in the r-th power of the maximal ideal of A. Thus we need to prove L e m m a 1. Let / }. Regard <2 as a set of prime ideals of I ® aJ. Then the intersection of all ideals in 2 is null.

Proof. Since I ®oJ is a finite extension of 0[[X,Y]] = A<§)0A, we may assume that I = J = A. Then A<8)0A can be identified as a measure space of WxW. Then, as seen in §3.6, the evaluation of power series O at (Pk,e,P/,ef) corresponds to the integration O h-> Jxke(x)y/e'(y)dji^(x,y). For fixed finite order characters e and e', the functions {xke(x)y/e'(y) I k > /} spans a dense subspace inside the space of continuous functions on WxW. In fact, by the variable change (x,y) h-> (xy'^y) = (s,t), this set equals the set of primes with respect to the variables (s,t) given by {(Pj i ee'-l,Pu t )lj>0 and / > 0 } , which corresponds to the space spanned by {s:'ee'"1(s)y/e'(y) I j > 0, / > 0}, which is obviously dense. Since the measure is determined by the dense subspace, we get the lemma, because (P contains a set of the above type for a suitable choice of e and e'. The above lemma shows the desired identity L p (^®(p c )p 0 = E»Lp(Xc<8>(p), because Lp(X®(pc)p0(P,Q) = (E»Lp(?ic®(p))(P,Q) implies

342

10: Three variable p-adic Rankin products Lp(?i(pc)p0 = E-Lp(Xc®(p) mod (P,Q)

for all (P,Q) e (P in the lemma inside the subring of Frac(l <8>oJ) *n which the denominators of Lp(A,®(pc)p0 and Lp(Xc®(p) in the field of fractions of I <§>oJ. That is, we have as a function of X(\)xX(J)

where Lp(X®q>c) is the function in Theorem 4.1 and Lp(kc®(p) is the function given in Theorem 7.4.2. Note that E = (1-7* —) is contained in I <§>oJ and hence holomorphic MT(p)) everywhere. When (p = X, it has a zero along the diagonal line (i.e. E is divisible by X-Y). On the other hand, the two variable L-function Lp(^c®A,) has a simple pole at the diagonal. Thus we know that Lp(A,®(pc)(P,Q,Po) does not have any singularity at the diagonal line. Since we can compute the value Lp(A,®(pc)(P,Q,Po) even for non-primitive X?, we can remove the condition (P3) in the evaluation formula in Theorem 7.4.2: Corollary 1. Let the notation be as in Theorem 10.4.1. Assume that G(Q) is primitive and %P" V Q ^ primitive. Then for the two variable L-f unction in Theorem 7.4.2, we have S(P)Lp(?tc<x)(p)(P,Q)

{

1 if Xp is primitive,

f, MT(p))c

otherwise.

Proof. We know the following formula by (1) when X? is primitive and by Theorem 4.1 (combined with the same computation which yields (1)) when X? is not primitive: S(P)Lpa®(pc)(P,Q,R) = (-l) / r(r+l)r(/+r)E(/+r)L(/+r,?ip®(poc)

10.6. Concluding remarks

343

Here note that L(s,Xp®(pQC) = Ei(s)L(s,?ipo®(pQC) for the primitive algebra homomorphism Xp° associated with X?, where

l(S) =

J 1 if X is p r i m i t i v e , |(l-?ip(T(p))-1(pQC(T(p))pk-1-s) otherwise.

We can also write, if X is primitive,

Then by the functional equation, we see that (27t)-2k+1+/r(k-/)r(k-1 )L(k-1 ,(V) c ®(p Q )

This shows that S(P)Lp(?t(pc)(P,Q,Po)

Then the result follows from Theorem 1.

§10.6. Concluding remarks In this final section, we try to give some indications for further reading. It is certainly affected by prejudice on the author's part and not at all exhaustive. In this book, we have discussed the theory of the critical values for L-functions of two algebraic groups GL(1) and GL(2) defined over the base field Q. About L-functions for more general algebraic groups and related subjects (especially "the theory of automorphic representations"), the reader may take a look at some articles in [CM] and [CT] and other articles and books quoted there. We have proved many algebraicity results for the critical values of L-functions in the sense of Deligne and Shimura [D]. The reader who is interested in such algebraicity results for L-functions of GL(2) and more general algebraic groups should consult Shimura's papers quoted in the references, especially [Shll, Shl2]. We should also note a paper of Blasius [B] on this subject, which gives a proof of a conjecture of Deligne about CM periods and the critical values of Hecke L-functions of CM fields. For the philosophical and geometric background of such critical values and their transcendental factors, we refer to [D] and some expository articles in [RSS] and [CT]. There exists a largely conjectural theory (due to Beilinson and others)

344

10: Three variable p-adic Rankin products

for L-values at integers outside the critical range, although we have not touched anything about that in the text. The reader may consult [RSS] and some expository papers in [CT] for these topics. We have also proved the existence of many p-adic L-functions out of the algebraicity results of these L-values. Such theory for GL(1) and GL(2) is now available for a large number of base fields, in particular, totally real fields and fields containing CM fields. As for abelian p-adic L-functions for CM fields, the reader will find every detail in an excellent article of Katz [K5], and for such p-adic L-functions with auxiliary conductor outside p, an exposition is given in [HT2]. As for p-adic L-functions for GL(2), some generalization of the results in Chapters 7 and 10 can be found in my paper [H8] and [Pa]. We have only discussed the automorphic side of the theory of p-adic L-functions in the text. As for the geometric side (i.e., the theory of motivic p-adic L-functions), see the articles of Coates and Greenberg in [CT] and articles quoted there. After having constructed p-adic L-functions, it is natural to ask their meaning. A partial answer to this question is supplied by the so-called "main conjectures" of an appropriate Iwasawa theory. As for the original Iwasawa conjecture proved by Mazur and Wiles ([MW] and [Wi2]), see the books of Washington [Wa] and Lang [L]. For the motivic (or geometric) side of such theory, see [Mz2] and [CT], [RSS]. For the automorphic side, see [MzS], [MTT], [Wi2], [MT] and [HT1-3]. Finally, as for the A-adic Galois representation described in §7.5, generalizations of the result in §7.5 to totally real base fields can be found in [Wil] and [H10]. The existence of such large Galois representations can be understood well from the perspective of deformation theory of Galois representations developed by Mazur [Mz3] (see also [HT3]).

Appendix: Summary of homology and cohomology theory In this appendix, we give a summary of the theory of cohomology and homology on complex and real manifolds, in particular, Riemann surfaces in order to make the text self contained. However we will not give detailed proofs for all of the material presented here. For sheaf cohomology, we refer to [Bd] and for group cohomology to [Bw] and for singular cohomology to [HiW] for details. Let X be a compact Riemann surface. We remove from X a set S of finitely many points and write Y for the resulting open Riemann surface. We fix a base point y e Y and let F be the fundamental group 7Cy(Y). Let R be a commutative ring and for any left R[F]-module M, we define the cohomology group H i (F,M)(=Ext ?m (R,M)) as follows. Let be an exact sequence of R[F]-modules, where on R, F acts trivially. When the Fi are all R[F]-free modules, we call F an R[F]-free (acyclic) resolution of R. Then by applying the contravariant functor HomR[p](*,M), we have a complex d\* 82* 0 -> HomR[pj(Fo, M) •-» Hom R[r j(Fi, M) -» HomR[rj(F2, M) -»•••. Then we define H°(r, M) = Ker(3i*) = HomR[r](R, M) = M r and H q (F,M)=Ker0 q + i*)/Im0 q *) for q > 0 . The independence of the cohomology group of the choice of the free resolution follows from Lemma A.I. If we have another resolution (which may not be R[F]-free), then there exists a morphism (p : F -» F' of complexes extending the identity on R and (p is unique up to homotopy equivalence. Proof. We can define inductively an R[F]-homomorphism F! extending the identity map on R, we now show the existence of an

346 RpTI-morphism

Appendix 5j : Fj_i —> F'j

such that

\|/ 'j-xj/j = 8J O 3J+3J + I°8J + I.

Since

e'oOj/'o-xj/o) = id-id = 0, Im(3'i) = Ker(e') z> Imty'o-Yo). Then picking any y a such that Vo(fa)"Vo(fa) = 3'i(ya), we define 5i(f a ) = y a for a basis fa of FQ and extend it by R[r]-li n earity to FQ. Thus \|/'o-\(/o = 3'i°5i. Suppose that we have 8i for i = l , . . . , j . Then from Vj-i-Yj-i = 8j-i°3j-i+3 j°5j, we see that ^ j°(vrVj-5j°aj) = d j°v'j-3 jo\|/j-a jogjoaj = 31jo\|/lj-31jo\|fj-(\|ftj.1-\|rj.1-8j.io3j.1)o3j = 0. This shows Im(3j+i) = Ker(3j) => Im(\|/j-\|/j-8jo3j). Taking a basis x a of Fj, we can find y a e F'j+i such that 3 j+i(ya) = (Yj-Vj-8j°3j)(xa), and we define 8j+i(xa) = y a and extend it to Fj by R[F]-linearity. Then we have V'j-Vj = 8jo3j+3fj+io8j+i. Thus \|/' is homotopy equivalent to \|/. Supposing F1 is also R[F]-free and interchanging the role of F and F , we find a morphism 9 ' : F —> F extending the identity on R. Making F = F' and replacing \\f by the identity map and \|/' by (p°(p\ we know that cp induces an isomorphism between the cohomology groups of F and F . Thus the cohomology group is independent of the choice of the resolution. We define a standard R[F]-free resolution of R as follows. Let F q be the tensor product of q+1 copies of R[F] over R and consider it as an R[F]-niodule via multiplication by R[F] of the first factor. Then F q is a free R[F]-module with a basis {[yi,...,y q ] = l®Yi®---®y q | (yi,..., yq) e H } . Then we define 3 q : F q -» Fq_i by 3i[y] = y-1 and for q > 1 3[Yi,...,Yq] =Yi[Y2, —.Yq]+XjLi (-^[Yi'—»YjYj+i.—>Yq]+(-l)q[Yi,....Yq-i] and extend it R[r]-linearly on F q . One can compute that 3q_i°3q = 0. Defining e : Fo = R[F] —> R by e(LySLyy) = Zyay, we see also that eod\ = 0. Noting that YtYl* "-> Yq] f° r (Y»Yi»--»Yq) G T q + 1 gives an R-basis of F q , we define an R-linear map D q : F q -> F q+ i by Dq(Y[Y2, ..., Yq]) = [%Yi. .... Yq]- T h e n w e s e e easily that 3qDq-i+Dq3q_i = id (q > 1) and 3iDo+e = id. This shows that F is an R|T]-free acyclic resolution of R. Let C = C(TM) be the space of functions on F 1 into M and put C°(r,M) = M. Note that any R[F]-linear map from F q to M is determined by its values on the standard basis [Yi» •••> Yq] a n d hence HomR[rj(Fq, M) = C q (r,M). Then the differential map

Appendix

347

3 : C 1 - > C 1 + 1 induced by d on F is given by 3u(y) = (y-l)u for U G M if i = 0, and if i > 0,

Then H ^ M ) = Z ^ F J v i y B ^ M ) where Z^FJVl) = Ker(3 : C1 -> C i+1 ) and B ^ M ) = Im(3 : C1"1 -> C1). Thus we again get H°(r,M) = M r = {x G M | yx = x for all y e F}. Any element in Z^I^M) (resp. B^FjM)) is called an i-cocycle (resp. an i-coboundary). In particular a 1-cocycle u : F - » M is a map satisfying u(y8) = u(y) + yu(5) for all y,8 G F and a 1-coboundary u is a map of the form u(y) = (y-l)x for any X G M independent of y. This shows that u ( l ) = 0 , u(y 1 ) = -y^uCy), uCySy"1) = u(y) + 711(8) - ySy^ufy) for cocycle u, and H ^ M ) = Hom(F,M) = Hom(Fab,M) if F acts trivially on M. Let H be the universal covering space of Y and n : H -> Y be the projection. For each S G S, we consider ns e F which corresponds to the path starting from y turning around the point s in the counterclockwise direction, and returning to y. Let r s = {7tsme r | m e Z}. We consider the set of all conjugates of 7CS for all s G S in F , which will be denoted by P. We define the parabolic cohomology group by H|>(r,M) = ZJ^(T,M)/B1(r,M), H|(F,M) = Z2(F,M)/B^(F,M), where zJ>(T,M) = {u e Zl(TM) I U(TC) G (TC-1)M for all KG P}, B|(F,M) = {3u I UG Cl(JTM) with U(TC) G (TC-1)M for all n G P}. Now the restriction of each 1-cocycle u from F to F^ = {7Cm|7CG Z} (n G P) yields a morphism res^: H^FjM) —> H^r^,]^). Therefore if u is a 1-cocycle of Tn, then u(7Cr) = (l+7C+7i;2+ ••• +7Cr"1)u(7c) and r u(7i" ) = -nMn1) = -7C"r(l+7C+7C2+ ••• +7ir4)u(7c) for r > 0 . Thus the cocycle u is determined by the value U(TC) at the generator 7C. If U(K) = (7i-l)x for some x G M, then u(7Cr) = (1+7C+TC2+ ••• +7ir"1)(7C-l)x = (TIM)X and

U(7fr) = -7fr(l+7C+7t2+ — +7Cr-1)(7l-l)x = (7C-r-l)x. Thus H ^ F S J M ) = M/(TC-1)M and we now know the exact sequence 0 -> HJ>(F,M) -> H 1 ^ ) -> line? H^F^M), where the last map is given by nTires^. Actually, if U(TI)G (TC-1)M, then for any

348

Appendix !

u(Y) = u(y)

V(Y) Thus, in fact, the conditions defining the parabolic cohomology group are finitely many, and we have 0 -> Hj,(r,M) -> H ^ M ) -> eserNpH 1 ^,]^). We now consider another description of Hp(r,M) by using a simplicial complex. Let Yo be an open Riemann surface obtained from X by excluding a small disk around each point s e S without overlaps. Let us take the pull back Ho of Yo to H, which is a simply connected open subset of H. We make a simplicial complex K with the underlying space Ho such that (Tl) (T2) (T3)

Every element of T induces a simplicial map of K onto itself, for each cusp s e S, the boundary of the excluded disk is the image of a 1-chain ts of K, there exists a fundamental domain of Oo in 9J§ whose closure consists of finitely many simplices in K.

We can construct such a complex by first taking a fundamental domain of <X>o and then making a finite simplicial complex £ with the property (T2) of the closure of the fundamental domain and finally shifting this simplex to cover all Ho by elements of P. We consider the chain complex (Ai,9,a) over R constructed from K; thus, Ai is a free R-module generated by i-chains of K. We have an exact sequence (by the simply-connectedness of Ho) 0 -> A 2 —^-» Ai —^-> A o —2-» R -> 0, where 3 is the usual boundary map and a(X z c(z)z) = S z c(z) e R. We sometimes identify S with the set of generators {TIS} of F s for s e S. We define Ai(M) = HomR[rj(Ai,M). Thus we have another complex 0 -> HomR[r](R,M) —-2-* A0(M) —^-> Ai(M) — ^ A2(M) -» 0, which may not be exact. Define Z^K.M) = Ker(3 : Ai(M) -> Ai+i(M)) and Bi(K,M) = im(a:A i .i(M)-^A i (M)) and ffCK.M) = Zi(K,M)/Bi(K,M). We further define Z|>(K,M) = {ue ZX(K,M) I u(ts) e (TTS-1)M for all S G S } , B^(K,M) = {3u | u(t s )e (7Cs-l)M for all S G S ) , HJp(K,M) = Zj>(K,M)/B1(K,M), Hp(K,M) = Proposition 1 (Shimura [Sh, Prop.8.1]). There is a canonical isomorphism

Appendix

349

Hi(K,M)=Hi(T,M), where "*" indicates P or f/ze wswa/ cohomology. Proof. We shall give two proofs of this fact. By construction, A : 0 -> A 2 —^—> Ai —^-> A o —^—> R -^ 0 gives an R[F]-free resolution of R and thus we can compute H^FJM) by using this resolution, which yields the above isomorphism for H1. For each n e P with n = y^sY"1, y(ts) is a simplex of K. Identify R with the universal covering space of the image of y(ts) in Oo and let i^ : R - » H be the induced map. Then the closure y(t s ) gives a triangulation of a fundamental domain of F^XR in i^R). Thus we can define Ai(rc) to be a free RIT^-module generated by i-simplices in y(t s ) and we have an RfFJ-free resolution: A(7i): 0 -> AI(TC) —^-> A0(7i) —-2-* R -> 0. We have a natural inclusion i^: A(%) -» A induced by i^. Thus we have i : ©TCepAfa) -> A. We have a natural action of F on ©^pAiCTi) which just permutes its simplices. Writing F = { F q } (resp. F(TC) = {Fq(7c)}) for the standard R[F]-free (resp. RfFJ-free) resolution of R, we have a commutative diagram,

i A

i >F,

where the vertical maps are the natural inclusions and the horizontal maps are the maps extending the identity on R as constructed in the beginning of this section. Then applying the functor HomR[rj(*, M) to this diagram and computing the cohomology, we have another commutative diagram, H 1 (F,M)^e s e sH 1 (F 7 r s ,M)

i

i

H1(K,M)-^©SGSH1(K(TCS),M),

where Hq(K(7i),M) is the cohomology group of HomR[r7I](A(7i), M). Since the vertical arrows are isomorphisms and the parabolic cohomology group is defined by the kernel of the horizontal maps, we see that Hp(K,M) = Hp(F,M). We now give another (more explicit) proof given in [Sh, §8.1]. We compute the cohomology group H^FjM) by the homogeneous chain complex; that is, we define (Q,3,a) as follows. For i > 0, Q is the R-free module generated by all the ordered sets (yo,yi, ..., Ji) of i+1 elements of F, the differential 3 : Q -» Q_i is defined by 3(Yo,yi,..., Yi)= X j = 0

350

Appendix

the augmentation a is given by a(Eiri(Yi)) = Siri and F acts on Q by Y(Yo>Yi> Yi).

Then

gives an R[F]-free resolution of R. In fact, we can easily check that this complex is isomorphic to the standard one F by the R[F] -linear isomorphism i given by i((Yo,Yi, •••> Yq)) = Yo[Yo"1YiJYi"1Y2, • ••, Yq-fVqL Thus by putting Ci(M) = HomR[T](Ci,M), the cohomology group of the complex 0 —2-* Co(M) —^-> Ci(M) —d—> ... gives the cohomology group H^I^M). As already remarked, we can define an isomorphism of complex (C(M),3) to (Q(M),3) by defining u' e Ci(M) out of u e C(M) by u'((Yo, Yi> ••., Yi)) = YouCYo'Vi, Y f S , . . . , (Yi-i)"SO. The isomorphism of H^FjM) onto H^KjM) can be constructed as follows. We take a finite set of i-simplices Si in K so that any simplex in K can be written uniquely as Y(S) f° r YG T. In particular, we can include ts in S\ and qs in So if 3ts = qs-7Cs(qs). Then we define a map fo : Ao -» Co by fo(Y(s)) = (Y) for each 0-simplex s in So, and then a°fo = a on Ao. Thus a(fo(3s)) = 0 for all s G Si. Because of the exactness of Ci —> Co -> R -> 0, we can find fi(s) in Ai so that 3fi(s) = fo(3s). We may assume fi(ts) = (l,7ts). Then we define for any ye F, fi(Ys) = 7fi(s) and extend this map R-linearly to the whole of Ai. Since we have defined fi so that if 3s = 0 for S E K, then 3fi(s) = 0 and we have 3fi(3s) = 0 for se S2. Now we can find f2(s) e C2 so that 3f2(s) = fi(3s) by the exactness of C2 -> Ci -> Co. Then by the R[F]-linearity, we extend f2 to an R[F]-linear map of A2 into C2. The morphism f=(fo,fi,f2) from (Ai,3,a) to (Ci,3,a) we have constructed satisfies (i) a°f0 = a, f°3 = 3°f and f°Y = Y°f f(>r Y G r » (ii) fi(ts) = (l,7Cs) for se S. By interchanging the role of (Aud) and (Q,3), we can similarly construct g : (Q,3,a) —> (Ai,3,a) satisfying (i') a°go = a, g°3 = 3°g and g°Y = Y°g f° r Y G r , (ii') gi((l>7is)) = ts + (7Cs-l)bs with 1-chain b s such that 3bs = qo-qs fora fixed 0-simplex qo in So independent of s e S. In fact, first fixing a 0-simplex qo, we define go((Y)) = Y(qo)- Then, by definition, a°go = a. Thus we can find gi((Y>8)) such that 3gi((Y.8» = goO(Y>8)) = go((5)-(Y)) = 5(qo)-Y(qo). Then we can extend gi to the whole space Ci by R[F]-linearity. In particular Note that 3ts = qs-7Cs(qs)- Thus by taking b s e A\ so

Appendix

351

that 3b s = qs-qo (this is possible since a(qs-qo) = 0). Then we have 3(t s +0r s -l)b s ) = qo-TCs(qo), and we may define gi((l,7Cs)) = ts+(7Cs-l)bs, because gi((l>TCs)) can be any element x in Ai such that 3x = go(3(l,7Cs)). The maps f and g induce morphisms f* : HJ>(T,M) -> HJ>(K,M) and g* : H1>(K,M) -> H ^ M ) . In fact, if u : F -> M is a 1-cocycle, the corresponding homogeneous chain u' is given by u'((l,y)) = u(y). In particular if U(TCS) = (ns-l)x, then f*u'(ts) = u'°f(ts) = uf((l,fts)) = (7is-l)x, and thus f* takes parabolic cohomology classes to parabolic cohomology classes. Similarly, if u e HomR^A^M) with u(ts) e (TCS-1)M, then g*u((l,7C,)) = u(ts + (7Cs-Dbs) e (7CS-1)M. Since (Q,3) and (Aj,3) are R[F]-free resolutions, and f and g induce the identity on R, f°g and g°f are homotopy equivalent to the identity. That is, there are R[F]-linear maps U : Q —» Ci+i and V : Ai —> Ai+i such that fog. id = 3U+U3 and g°f - id = 3V+V9. The map U can be defined as follows. Since f(g((y))) = (y), we have fo°go = id, and we simply put Uo = UI Co = 0- We have 3(f(g(d,y)))-(i,y)) = f(gOd,y)))-3(i,y) = o because fo°go = id. Thus we can find U((l,y)) so that Then we extend U by R|T]-linearity to Ci. By definition, we have fi°gi - id = 3U+U3. Similarly, we see that a(f2(g2((l,y,5)))-(l,y,5)-UO(l,y,5))) = fi(giO(l,y,8)))-3(l,y,8)-3UO(l,y,8)) = 0 because 3U((5,y)) = f(g((5,y)))-(8,y) and 3(l,y,8) = (y,8)-(l,8)+(l,y). Thus we can find U(l,y,8) so that 3U(l,y,8) = f2(g2((l,y,8)))-(l,y,8)-UO(l,y,8)). Then we have U satisfying f°g - id = 3U+U3. In this way, we continue to define U inductively (actually, it is sufficient to have U defined as above because Hp(F,M) = 0 if i > 2). As for V, we proceed as follows. Since a(g(f(s))-s) = 0 by definition, we can find V : Ao -> Ai so that 3V(s) = g(f(s))-s for all s e Ao. Then we consider 3(gi(fi(s))-s-V(3s)) = go(fo(3s))-3s-3V(3s) = 0 by the definition of V. Now we can define V(s) for s e Ai so that 3V(s) = gi(fi(s))-s-V(3s) and continue to define V inductively. Then for each 1-cocycle u e Zi(M), we see that u°f°g-u = u3U+uU3 = 9uU e Bi(M). Thus g*f* = id and similarly f*g* = id on the cohomology groups. As already seen, they preserve parabolic classes and hence induce isomorphisms on parabolic cohomology groups.

352

Appendix

Let So be a subset of S and T be the disjoint union of the image of ts in Y for s G So. Let Kj be the subcomplex of K generated by all translations of ts by T. We put KT = K/KT. Consider the free R-module A?1 (resp. AT,i) generated by the i-simplex of KT (resp. KT). Then we write H§ (F, M) for the cohomology group of HomR[r](A^,M). When S = So, we write Hq (F,M) for Proposition 2 (Boundary exact sequence). We have a long exact sequence 0 -> H°SQ (T, M) -> H°(T, M) -> 0 SG s o H O (r7v M) -> H ^ (F, M)

-> H^r, M) - » e s e s o H 1 ^ , M) -> njo (r, M) -» H2(r, M) -> o. / * particular,

B°SQ (F, M) = 0 z/ So * 0 , and H* (r, M) = H | ( F , M).

Proof. We need the following well known fact: (1) If 0 ^ A - » B — > C — » 0 is an exact sequence of complexes, then we have a long exact sequence ••—> Hq(A) -> KP(B) -> Hq(C) -> Hq+1(A) -> HP+1(B) ->•••. This is checked by applying the snake lemma to the commutative diagram: Ap/9p-i(Ap.i) -> Bp/ap.i(Bp_i) -> Cp/ap.i(Cp.i) -> 0 (exact)

id

id

id

0 -> Ker(a p + i | A p + 1 ) -> Ker(3 p+ i | B p + 1 ) -> Ker(3 p+1 1 C p + 1 )

(exact).

We have an exact sequence 0 —» Ay —> A —> A —> 0. Since A T is also R[F]-free, this sequence splits as R[F]-module, and we have another exact sequence 0 —> Hom R [ r ](A T , M) -> HoniR[r](A, M) - ^ HomR [r](AT, M) -^ 0.

Then applying (1), we get the long exact sequence. The vanishing of H s (F, M) (when So ?* 0 ) follows from the injectivity of M r = H°(F, M) -+ S s e S o H 0 ^ , M) = 0 s e s o M r 7 t s . The last assertion follows directly from the definition .of H p (F, M). We now define the notion of sheaves on a smooth manifold Z of real dimension n. Here the word "smooth" means that Z is of C~-class. Let O(Z) be the category of all open subsets of Z; that is, objects of O(Z) are open subsets of Z and Homo(Z)(U,V) is either the inclusion map U —> V or empty according as U is contained in V or not. A presheaf F on Z is a contravariant functor on O(Z) having values in the category of abelian groups. Thus for each open set U in Z, F(U) is an abelian group and if U 3 V, we have a natural restriction map res u / v : F(U) -> F(V) satisfying resu/u = id and res v/w cres u/v = res u / w if U z> V D W . A presheaf is called a sheaf if the following axiom is satisfied:

Appendix (S)

If for

a given open covering

353 U = UiVi,

s i e F(Vi)

satisfies

res V i / V i n v .(si) = res Vj/VinVj (Sj) for all i and j with ViflVj * 0 , we have a unique element s e F(U) such that res u/Vi (s) = S{. If U = U[=1 Vi is an open covering, adding Vo = U to this covering, the condition (S) implies that s e F(U) is uniquely determined by res u/Vi (s) for all i = 1, ..., r. A morphism (]) of a sheaf F into another G is defined to be a morphism of contravariant functors. That is, for each open set U, we have a morphism (|)(U) : F(U) - ^ G(U),

and for every open subset

V

of

U, the

following diagram is commutative: 4>(U) F(U) -> G(U) 4/ resy/y

i> reSy^y

F(V) -> G(V). 4>(V) If U is an open subset of Z, then O(U) is a subcategory of O(Z). Thus we can restrict the sheaf to O(U). The restriction of F to U will be written as F l u If 71 : T —> Z is a surjective morphism of smooth manifolds, we can define a sheaf associated with T by T(U) = {s : U —> T I s is continuous and izos = id on U } . It is easy to verify the condition (S) for the usual restriction map r e s u / v ( s ) = s | v if U D V. Let A be any abelian group with the discrete topology. We consider T = ZxA with the product topology. Then if an open set U in Z is connected, any continuous section on U of K : T —> Z is a constant function, and hence T(U) = A. This sheaf T is called the constant sheaf A. A sheaf on Z is called locally constant if for each point z e Z, we can find an open neighborhood U of z such that F | u is a constant sheaf. Returning to the situation in Lemma 1, we can give plenty of examples of such locally constant sheaves. For any F-module M, we can define T = F \ H x M letting F act on H x M by y(z,m) = (7(z),ym). We put the discrete topology on M and put the quotient topology on T. Then the projection n : T —> Y = F \H gives a sheaf T, which we write ML For a small open set Uo of H such that yUoflUo * 0 <=> y = 1, writing U for the image of Uo in Y, we see easily that 7C"1(U) = UxM and hence M l u is a constant sheaf M. For every point Z E Y , we can always find such a neighborhood U, and hence M is a locally constant sheaf. Now we introduce the Cech cohomology group of a presheaf F. For an open covering Zl = {Ua}aGi* Z = UaeiUa and a = (ao,...,a r ) e I r + 1 , we write

354

Appendix

Ua = UaorV*riU(Xr. Then we define C ^ F ) s :I

r+1

to be a module of functions

-> IJaGiF(Ua) satisfying s(a) e F(U a ). Then we define

by, for a = (ao,...,a r + i)e I r+2 , (3s)(a) = Xj=o

a(j)

where a® = {oq e a | i * j} and we understand F(0) = 0. For example, d(s)aPy = sPy I Uap^rSay I UapY+Sap I Uapy w n e n r = h and when 3(s)ap = sp I U a p-s a I U a p, 3(3s)apy = dspy I U a p r 3s a y I Uapyf 3sa(51 U a p y = Sy I U a p r Sp I Uapy- { Sy I U a p r Sa I Ua(3y} +Sp I U a pyS a | U a p y = 0. This shows d2°d\ = 0. We can similarly check 3 r+ io3 r = 0 for general r. We write the cohomology group H q (C(^;F)) as Hq(Z;1i;F): H q (Z;^;F) =Ker(3 r )/Im(a r4 ). Another open covering 1^= {Vp}p€ j is called a refinement of U if there exists a map p : J —> I such that UpQ 3 Vj for all j . In this case, we write U > V, We define U and V to be equivalent if U > V and V> U Let C be the set of equivalence classes of all coverings of Z (C is a set because the set of all coverings is a subset of the power set of the power set of Z). The partial ordering ">" induces a filtered ordering on C In fact, if Zl and
Lemma 2. The morphism p: J -> I.

p* does not depend on the choice of the map

Sketch of a proof. Let I be another map from J into I having the same property as p. We show that p* and I* are homotopy equivalent. Let 5 : CqCU;F) -» C q+1 (^,F) be a map defined by (8S)(P) = Then by a direct computation, we get i-p = 63+38. By Lemma 2, if U> V, we have the canonical homomorphism

Appendix

355

) : Hq(Z;?i;F) If 1/ and V are equivalent, by the uniqueness i(Zl,ty°i('V,ZL) and i(y, V> <W, then 1(^,^)01(11,^ = 1(11,14/). Since C is filtered, we have an injective system { H q (Z; ?i;F),iCU,
Giving a 0-cocycle s in C°(t/;F) is tantamount to giving Si G F(Ui) for each i G I such that resui/uij(si) = resuj/uij(Sj). Thus if F is a sheaf, we have a unique s G F(Z) such that resz/ui(s) = Si. This shows (2)

H°(Z;
Proposition 3. Suppose that Z is simply connected and ^i={Ui}i e i is a covering of Z with a countable I such that (*) U a for each a G I r+1 w ezY/zer empty or simply connected for all r G N. //* resu/v is surjective for all V c U with V connected (this assumption holds if F is constant), then Hq(Z;£/;F) = 0

Proof. We fix (5e I and write U = Up. We consider UI u = {UnU a I ae 1}. Then we can define, for the fixed p G I

F | U and

8 r : CT+l( 0) in Cr+1(1/;F), we can find t such that s I u = 3t. The cochain t is uniquely determined by s modulo SCC'ZilujFlu)- By the same argument applied to U' = Up with UflU1 * 0 , we find t' so that 3t' = s | u1 on U'. Modifying t1 by an element in 3Cr(*Zi | u;F I \j), we may assume that resu/unu't = resuyunu't', because the restriction map 3C r (^lu;Flu)->9C r (^/lunu i ;Flunu 1 ) is surjective. Thus t extends to UJU1. By continuing this process, by the simple connectedness of Z, we can extend t to UiUi = Z. This shows the result. For a presheaf F, we study the presheaf F # : U H H°(U,F | U). The restriction map resu/v of F induces that of F#. By (2), if F is a sheaf, then F* = F. Let ^ = (Uihei be an open covering of U. To each s e F(U), assigning the 0-cocycle s(i) = resu/u/s) e C°(T/;F), we have acohomology class [s] in F#(U). Thus we have a natural morphism of presheaves i : F —> F # . If s G F^CU) and if

356

Appendix

resu/Ui(s) = 0 for all i, s itself vanishes because the image of s in H 0 (U;^;F|u) is 0. Thus s e F*(U) is determined by its local data. Let si for all i G I be sections in F*(Ui) satisfying resu/Ui (Si) =resu/Ui-Csj)^ or a ^ * anc^ j . Then we take a sufficiently fine open covering U\ = {l^lke J of Ui and choose a 0-cocycle s1 e C°(Zli;F\u) representing sj. Then writing p : IxJ —> I for the projection, we regard <]/= {l4)(i,k)eixJ as a refinement of 11. Then the cochain: (i,k) H> s\k) is a 0-cocycle in C°(V ;F I u)- Writing s for the cohomology class of this cocycle, we have plainly S[ = resu/Ui(s) for all i e I. Thus F* is a sheaf. If $ : F -» G is a morphism of presheaves and if G is a sheaf, <>| induces 0* : F* —> G# = G. It is obvious that $ = ( ^ i and (|>* is characterized by this property because any s e F#(U) can be described by local sections. Thus F # satisfies the following universal property: (3)

for each morphism 0 of presheaves from F into a sheaf G, there exists a unique <|>* : F* —» G satisfying 0 = (J^i.

By this universality, F* is uniquely determined by F (up to isomorphisms). We call F # the sheaf generated by F. We say a sheaf (resp. a presheaf) is a sheaf (resp. a presheaf) of R-modules for a commutative ring R if F(U) is an R-module for every open set U and resu/v • F(U) —» F(V) is an R-linear map if U z> V. For any presheaf F of R-modules, the sheaf F # generated by F is naturally a sheaf of R-modules. If F and G are two sheaves of R-modules, then we write F ® R G for the sheaf generated by the presheaf: U H F(U)<E>RG(U). The map (x,y) i-» x®y composed with the canonical map i is again denoted by the same symbol: (x,y) h-> x®y for (x,y) e F(U)xG(U). It is easy to verify the following universal property (see the description after Corollary 1.1.1): (4) If there is a morphism of sheaves of R-modules (|) : FxG —» H such that (U)(x,A,y) = (|>(U)(Ax,y) for all XeR, then there exists a unique morphism of sheaves of R-modules <(>* : F ® R G —> H such that <|>*(U)(xy) = <|)(x,y). If F is a subsheaf of a sheaf G, we can define a presheaf U \-> F(U)\G(U). We write FXG for the sheaf generated by the above presheaf. We have a natural morphism G -> G/H. On the other hand, it is plain from the sheaf axiom (S) that, if (p : F —> G is a morphism of sheaves, Ker(cp): U -> Ker(cp(U)) is again a sheaf.

Appendix

357

A sequence of morphisms of sheaves F — — > G — ^ H is called exact if Im(a) # = Ker(P), where Im(a) # is the sheaf generated by the presheaf U H> Im(a(U)). For any presheaf F and x e Z, we define the stalk F x at x by F x = limF(U), where the transition map is given by res u / v , and the order on the set of open sets containing x is given by the inclusion relation. Then it is easy to check from the definition that (5a) F # x = F x for all X G Z , and F —> G -» H is exact as sheaves if and only if IlxezFx -> IIxezGx -» IlxezH x is exact as abelian groups. In other words, (5b) if a : F —»G is a surjective morphism of sheaves, then for each g e G(Z), there exists an open covering {Ui)iei with fi e F(Ui) such that oc(fi) = resz/Ui(g) for all i. For each section f e F(U), we define Supp(f) to be the closed set in U defined by ( x e U | fx * 0}, where fx is the natural image of f in F x . Now we return to the original situation: X is a compact Riemann surface, Y = X-S and F is the fundamental group of Y. Let M be a F-module and M denote the locally constant sheaf on Y associated to M. Theorem 1. Let U = {Ui}iGi be a covering of Y with a countable index set I such that (i) for each Ui, there exists a simply connected open subset Ui* in H such that the projection n : H —» F\H induces an isomorphism Ui* = Ui, and (ii)for all r, U a * = U ao *lT • -flU^* is either simply connected or empty for all a G F +1 . Then there is a canonical isomorphism Hq(F,M) = H q (Y;^;M). Proof. We consider the open covering 11* = {y(Ui*)}yeT,iei °f H. By (i) and (ii), if U a * 0 for a = (ao,.-.,a r+ i) e I r+1 , then there exists a unique r+1 (Yo,...,yr)e T so that U a * = yoU a o*n-"ny r U ar * * 0 . We put U a * = 0 if U a = 0 . Then yU a * for y e F is either empty or simply connected. Thus if yU a * * 0 , then F(yUa*) = M. We consider the subset Ir in I r+1 consisting of a with non-empty U a . Let R[F][Ir] be the formal free module generated over R[F] by elements of Ir, and write [y,a] for the element (y,a) (ye F and a e I r )in R[F][U-Then we see that {[y.alJ^r.ae^ is a basis of R[F][IJ over R, and we have HomR(R[F][Ir],M) = Cr(^*;M) by c> — i>s where s(y,cc) = (J)([y,a]) e F(yUa*) = M. By this isomorphism, C(ZI*;M) has a natural structure as R[F]-module. We define an R-linear map

358

Appendix by 3[y,a] =

^=0

where a® = (ao,...,ocj_i,aj+i,...,ar). Obviously 3 r is R[r]-linear. Thus we have a complex R[ where Then C(^*;M) = HomR(R|T][r|, M) as complexes. Since H is simply connected, by Proposition 3, C(£/*;M) is exact and thus R[r][I] is an R[rj-free resolution of R. It is obvious that HomR[ r](R[r][U], M) = C(*Z;M) as complexes. Thus by Lemma 1, we get the desired isomorphism. Since we can take a simply connected polygon as the fundamental domain of Y, for any covering 0. /•••\ ot. TTr/r»/r* #-<\ f C if r = 0 and 1,

Cm) show H r (R/z,o={

0

otherwise;

Let F be a locally constant sheaf on Z having values in the category of finite dimensional vector spaces over C. We consider the sheaf $tpT (on Z) of smooth differential forms of degree r with values in F. Thus .%r(U) is the space of C°°-r-forms defined on the open set U with values in F(U); so, -#Fr = -#cr®cF in the sense of the tensor product of sheaves (4). Since the exterior derivation d is defined locally on J%c and locally .%r(U) = J3cr®F(U), d is well defined on %$ and we have a resolution of the sheaf F: The above sequence is exact in the sense of sheaves (5a,b) by Poincare's lemma, which tells us the validity of (5b) for dq : ^tpq ~> ^q+i = Ker(dq+i) on every simply connected open set. Let U - {Ui}i€i be an open covering of Z. Let {(j)j}je j be a partition of unity subordinate to the covering U Thus there is a map a : J —> I such that ty is a smooth function on Z with Ua(j) => Supp((|)j), all but a finite number of (|)j vanish at each point x € Z and Xjj(x) = 1 fc>r all

Appendix

359

x e Z. Such a partition of unity is known to exist. Then we put for all cocycles s e C(U;J^P), s'(P) = £j(|>js(o(j)UP) for p e Ir. Then k=0

j

j

k=0

Since 3s(aG)LJa) = s(a)-Ifc =0 (-l)Va(j)Ua (k) ) = 0, we see that This shows that (6)

H q (Z;
Let H^ R (Z, F) be the cohomology group of the complex

MZ): 0 -* ^°(Z)^U V ( Z ) - ^ F 2 ( Z ) - ^ U - . The group H^ R (Z, F) is called the de Rham cohomology group. Then we have Theorem 2. If the covering

U of Z satisfies the condition of Proposition

3,

q

then there is a natural isomorphism

H£ R (Z, F) = H (Z;
Proof. For any sheaf, it is easy to see that the presheaf: U H> C(1l | u;F I u) = 0 q C q ( ft I u;F I u) is actually a sheaf of complexes, where U \ u = {UiflU l i e I } . We still write this sheaf of complexes as C(ft;F). From the exact sequence:

we get another exact sequence of sheaves of complexes: (*)

0 -> C(ft;^) -> C(ft;^pq) ~> C(ft;Vi) ^ 0.

In fact, for each simply connected open set U, Poincare's lemma tells us the surjectivity of d : .%q(U) —» 2^+i(U). Since every non-empty U a (a e f +1 ) is simply connected, (*) is not only exact as sheaves but also exact as complex of C-vector-spaces. Then we have the long exact sequence (1) attached to (*):

0

H

^

Since the restriction map of ^Fq(U) to ^pq(V) for V c U is surjective, we can apply Proposition 3 and know that Hp(Z;ft;.%q) = 0 for all p > 0. This shows Hq(Z;1/;F) = H\ ^

360

Appendix

A sheaf F is called flabby if for every inclusion of open sets V c U, res u / v : F(U) -> F(V) is surjective. We define abelian groups F(F) and FC(F) by F(F) = F(Z) and F c (F)={fe F(F)|Supp(f) is compact}. Lemma 3. If 0 —> F -> G -> H -» 0 is an exact sequence of sheaves on Z. Suppose that F is flabby. Then 0 -> F(F) -> F(G) ->F(H) ->0 and 0-> FC(F) —» FC(G) —» FC(H) —> 0 are exacf sequences of abelian groups. In particular, if F attd G are flabby, then H is flabby. Proof. We only need to check surjectivity of a : FC(G) —> FC(H) and F(G) -> F(H). Let s e F(H). Let C be the collection of all pairs (t,U) (t€ G(U)) such that cc(U)(t) = resz/u(s). We give an order on C so that (t,U) > (t',Uf) if U D U 1 and res^Ct) = t'. Evidently C is inductively ordered. Thus there is a maximal element (U,t) in C by Zorn's lemma. Suppose U ^ z . Let x G Z-U and take a small neighborhood V in Z of x. If V is sufficiently small, we can find v e G(V) SO that (V,v) e C by the surjectivity (see (5b)). Then res u/u p V (t)-res v/V p U (v) e Ker(a)(UflV). Thus by flabbiness of F, we can find f e F(ULJV) such that res

uuv/unvff) = res u/unvW" res v^/nu( y )-

This implies that t and f+v coincide on UflV. Then by the sheaf axiom, we can find t' e G(ULJV) such that res

uuv/v( tf ) = f+v

a n d res

uuv/u(tf) =

t

Then (ULJV, t') is larger than (U,t) contradicting the maximality of (U,t). Thus U = Z and oc(t) = s. This shows the assertion for F. When s e FC(H), then V = Z-Supp(s) is an open set and res z/v (s) = resz/v(oc(t)) = 0. Thus we can find t' e F(F) such that Supp(t-t') c Supp(s). Then t-t1 e FC(G) and cc(t-t') = s. This shows the assertion for F c . Let F be an arbitrary sheaf. Let Fl: 0 ->F -> Fl0 -> Hi -> Fl2 -^ ••• be an exact sequence of sheaves where the Fli are all flabby. Such a complex is called a flabby (acyclic) resolution. There is a standard flabby resolution F1(F) of F: We define F1°(F)(U) = n x euF x . Then F1°(F) is plainly a flabby sheaf. Then the diagonal map: fh-> Ilxeufx takes injectively F into F1°(F) by (S). After defining Flk(F) and 3 k _i: F\kA(F) -^Fl k (F), we just define Flk+1(F) = Fl°(Coker(3k_i)) and ^ : R k (F) -» Coker(3k_i) -> Flk+1(F), where the last arrow is the natural diagonal map. We define the sheaf cohomology group Hq(Z,F) (resp. the compactly supported sheaf cohomology group

Appendix H q (Z,F))

to be the cohomology group of the complex

361 F(F1(F)) (resp.

r c (Fl(F))). Proposition 4. There exists a canonical isomorphism Hq(Z,F) = Hq(Z;F). Proof. By Lemma 3, Hq(Z,F) = H q (Z,F) = 0 if F is flabby. First suppose that F is flabby. Then the sheaf Cq(^i;F) is flabby. Consider the exact sequence of sheaves 0 -» Z^ -» Cq(<£i;F) -» Z^+i -» 0 for Zq = Ker(3q). This induces an exact sequences of complexes 0 -> H(j^) -> F1(C(^;F)) -> Fl(2^+i) -» 0. Note that F ^ C ^ ^ F ) ) is flabby. Applying (1), we have a long exact sequence: H p (Z,C q (^;F)) -> Hp(Z,z;q+1) -> H ^ Z , ^ ) -> H ^ ^ ^ C ^ ^ F ) ) . Since both ends of the above sequence vanish because of flabbiness of we have This shows 0 = Hq(Z,Zo) = H^CZ,^) = • • • s H^Z^q.i) = H q (Z; ^;F). Thus we see that (7) If F is flabby, then Hq(Z;ft;F) = 0 for all covering U Now we treat the general case. Consider the exact sequence of sheaves: 0 -> Z'q -> Flq(F) -> Z'q+i -> 0 for Z'q = Ker(3q). Note that Flq(F) is flabby. From the long exact sequence of Cech cohomology, we have an exact sequence 0 = Hq(Z,Flq(F)) -> H q (Z,Z q+ i) -> H q+1 (Z,^ q ) -> Hq+1(Z,Flq(F)) = 0. Since both ends of the above sequence vanish because of flabbiness of C(1i;F), we have H q (Z,£ q + i) = H q+1 (Z,Z q ). This shows H q (Z,F) = Hq(Z,Z'o) £ Uq-\Z,Z'i) = — = ttl(Z,Zq_i) = H^ZJF). In fact, we can compute the sheaf cohomology group Hq(Z,F) of F using any flabby resolution 0 —> F —> Fl of F. Out of the sheaf exact sequence 0 -> Z^ -> Flq -> Z^i -> 0 for Zq = Ker(3q), we have another exact sequence of complexes 0 -> Fl(Z^) -> Fl(R q ) -> H(2^ + i) -> 0. Applying (1) to this, we have, for H* denoting any one of H q and H q , Hq(Z,F) = Hq(Z,2o) = H r ^ Z , ^ ) £ - £ H2(Z f ^) = H q (Fl). Thus we have (8)

For any flabby resolution 0 -> F -> Fl, Hq(Fl) £ Hq(Z,F) £ Hq(Z,F) and H q (R) = Hq(Z,F).

362

Appendix

Suppose F is a presheaf with surjective res u / v for every inclusion V c U and F satisfies a part of (S): (S!)

If for each covering {Ui}, a given set of elements Si e F(Ui) satisfies resz/uiJUjSi = reszyujJUjSj* there exists s e F(U) such that resz/u^s) = Si for all i;

then we see easily that F # is flabby. Now assume that Z has a triangulation K. We now introduce the subdivision process of complexes. Let A2 = (a,b,c) be an r-simplex. We define a subdivision Sd(A2) of A2 by all the simplices in Figure (ii): c1

b1

c

c

Figure (i)

Figure (ii)

Here h is the barycenter of A. Thus writing h*(xi,...,xr) = (h,xi,...,xr), we see that Sd(a) = a, Sd(b,c) = (b,a')+(a',c) for the barycenter a' of the line segment (b,c) and Sd(A) = h*Sd(3A). By the last formula, we can inductively define the subdivision operator Sd for all r-simplices. We apply this process of subdivision to each simplex of the complex K = Ko. We write the resulting simplex as Ki = Sd(Ko). We continue this process n times and write the n-th subdivision obtained by Kn = Sdn(K) = Sd(Kn_i). Finally we define Koo = limK a . Let R be a commutative algebra with identity. For any open set U on Z, we can consider a module Aq(U) of formal linear combinations £aA€RaAA f° r aA e R for i-simplices A in KeoflU. Then we have a covariant functor A q : U h-> Aq(U) with natural inclusion map Incu/v * Aq(V) —» Aq(U) if U 3 V. Thus we have a presheaf Hom R (A q ,F): U H> HomR(Aq(U), F(U)). If F is locally constant, HorriR(Aq,F) satisfies (S) and hence it is a sheaf and HoiriR(Aq,F) gives a flabby resolution of F. We can thus compute Hq(Z,F) and Hq(Z,F) using HomR(Aq,F). We now return to the situation in Proposition 1: Z = Ho and F = M. Let Aq>a be the R-free module generated by q-simplices in K a and put S q (K a ;M)=Hom R (Ai, a ;M) with T-action (ft | y)(A) = Then we define H q (K a ,M) by the cohomology group of the complex

Appendix

363

0 -> S°(K a ;M) r -» S^K^M) 1 " -* S 2 (K a ;M) r -» 0. We want to show that Sd induces an isomorphism H q (K a ,M) = HFQKa+i.M). Let ^a be the triangulation of Yo induced by K a . Each vertex x e £ a + i is a barycenter of a unique simplex A of £ a by the definition of the subdivision. Choose one vertex y of A and define cp(x) = y. We know that if (x,y) is a 1-simplexof fat+i, then either x or y, say x, is the barycenter of itself. Hence y is the barycenter of (x,z). Then cp(x) = x and either (p(y) = x or (p(y) = z. In either case (cp(x),(p(y)) is a simplex of £ a + i . Here we allow repetition of vertices: (x,x) implies the 0-simplex x. Similarly, if (x,y,z) is a 2-simplex of £a + i, then looking at figure (ii), we may assume that x is the barycenter of x itself, y is the barycenter of (x,a) e £ a and z is the barycenter of (x,a,b) e kjx. Thus we define cp(x) = x, cp(y) = x or z and (p(z) = x, a or b. In any choice, (cp(x),(p(y),(p(z)) is a simplex of £a- Thus 9 induces a simplicial map and hence a continuous map on Yo into itself, which we denote by the same letter (p. We want to show that (p is homotopically equivalent to the identity. For each x e Yo, by definition, both (p(x) and x belong to the closure of a 2-simplex t(x). Moving Yo into R 2 by the homeomorphism given by the triangulation £, we may assume that Yo is embedded in R . Then we define F : [0,l]xY 0 -» Y0 by F(u,x) = (l-u)(x-a)+u((p(x)-a)+a if t(x) = (a,b,c) for three vectors a, b and c in R 2 . Obviously F is continuous and hence gives the homotopy equivalence between (p and the identity. Extending the simplicial map cp to (p : Aqj(X+i —» Aq>a by R[F]-linearity, we have an R[F]-morphism (p which induces an isomorphism (p* : H q (K a + b M) = H q (K a ,M) satisfying (p*oSd = Sdocp* = id. We now identify H q (K a , M) with Hq(K, M) by Sd a . Since Y o is isomorphic to Y as a smooth manifold, we can compactify Y identifying Y with the closure of Yo in X. We write Y s for this compactification. Note that Y s and X are different; that is, X has one point at the cusp but the boundary of Y s at the cusp is a circle S 1 isomorphic to a small circle around the cusp in X. We can think of something in between Y and Y s depending on So for a given subset So of S. We remove the boundary at s e So from Y s and write it as Y s - S °. Note here that Y 0 = Y. Since Sq(Koo;M)r are the global sections of the sheaf HomR(Aq,M) on Yo, we have Proposition 5. Let the notation be as in Propositions 1 and 2 and Theorem L Then we have canonical isomorphisms H q (Y s - s °,M) = H^o(F,M) for all subsets S o in S and Hq(Y,M) = Hq(K,M). Then by the boundary exact sequence in Proposition 2, we have

364

Appendix

Corollary 2. Let the notation be as in Proposition 2. We have a long exact sequence 0 -> H°(YS"S°,M) -» H°(Y,M) -> 0 se s o H o O s Y*,M) -> ^(Y S - S °,M) -> H^YJMD -> e s ^ o H ^ a s Y * ^ ) -> if (Ys"So,M) -> H2(Y,M) -» 0, 3SY* is the boundary around s. We now briefly recall the Poincare duality for Y. Our proof of the duality is merely a sketch, and the reader should consult standard texts (for example [HiW, 4.4.13]) for details. Suppose that R is a field and M is a finite dimensional vector space over R. Let <>| e HomR(Ao)Ct>M*) be a cocycle for M* = HoiriR(M,R), where a is a fixed integer (which we make large if necessary). We suppose that <|) is compactly supported on Y. Then we can find a small open neighborhood of 3YS which does not meet the support of $. We take a cocycle COG HorriR(A2,oJvI). Then we define for each i-simplex A of fox, <(J),co)(A) = <<|>(A),©(A)> for the dual pairing ( , ) : M*xM -> R. Since A is simply connected, <>| is a constant function on A. Here we write <|>(A) for this value of ty on A. It is obvious that ((|),co) e HomR(A2,a,R) and Supp(((),co) is compact in Y. We see easily that ((|),3co) = 3(<|),co) and hence (<]),co) is a 2-cocycle. It is well known (see Proposition 6.1.1 for a proof) that H 2 (Y,R) = R. That is, we have a pairing (,):H°(Y,M*)xH^(Y,M)->R. Suppose (0,0)) = dr[§ for all $ with compactly supported r^ e HorriR(Ai,R) depending on 0. We fix a basis {ei, ..., er} of M and take its dual basis {ei*,..., er*} of M*. We write <>| = Z i t e * and co = ZiO)iei. Then we can define a 1-chain ^ e HomR(A1,M) by £i(Ai) = (n0ie.*(Ai)ei* on each 1-simplex Ai in the A. Put S; = £ & . Then we see from construction that 3^ = co on A. We extend this process of defining £ to 2-simplices adjacent to A. Then inductively, we get £ such that 3 ^ = co. Thus the pairing is non-degenerate on HC(Y,MJ. Similarly, we can show that the pairing is non-degenerate on H (Y,M*) much more easily because there is no-restriction for co to be a 2-cocycle (i.e. Y is two dimensional). Thus we have Proposition 6. Suppose that R is afield and M is finite dimensional. Then the pairing ( , ) : H°(Y,M*)xH^(Y,M) -> R is a perfect duality.

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Answers to selected exercises §1.2. 1. We only need to show that (2nl(m):&+(m)) is finite, because I(m)lTC\I(m) injects into I/(P. We see by definition that the natural map 0-{O} —»fP, which takes each integer a e O to the principal ideal generated by a, induces a surjection from the finite group (O/m)x onto #\l(m)l R r xC t given by F 3 a h-» ( a ^ ) induces an injection of 2(m)/2+(m) -> (R x )7(R>o) r , where R > 0 = {x e R | x > 0}. Thus (2(m):(P+(m)) < 2 r . This shows the desired assertion. 2. (b) What we need to show is that the integer ring O is not a unique factorization domain. Note that 0={x+yV-5 | x,y e Z}. We have a decomposition 3-7 = 21 = ( 1 + 2 V - 5 ) ( 1 - 2 - N / - 5 ) . All these factors cannot be factored into a product of two non-units. For example, if 3 = a(3 with non-unit a and (3, then 9 = N(a)N($) and hence Af(oc) = 3 because otherwise (3 becomes a unit. This is impossible because 5 mod 3 is not a square. Similarly, one can show that 7, (1+2V^5) and (l'2^f^5) are all primes. Obviously, (l±2V~5)/3 and (l±2V~5)/7 are not integers, and hence the prime factorization in O is not unique. 3.

By the Minkowski estimate, there is

1 < | iV(a) | < | V W l ^ .

0* a e O

such that

Thus 1 < ^ - j - < | VE>F I. There is another way

to prove this only using an argument similar to the proof of Lemma 1.2.4: In fact, by the lemma, if Ki Kd = I VDp I, then the set ..,K d ) = { 0 * a e

o\

| a ( i ) | < K{ for all i}

is not empty. In particular, for any 8 > 1, choosing a small e > 0 so that 8K!.(K 2 -e)

(Kd-e) = | VD^ I, then Y(8Ki,(K 2 -e),...,(K d -e)) * 0 . By putting

X(Ki,...,K d ) = {0 * a G O

| a ( i ) | < Ki for all i > l

and

I oc(i) I < K i } ,

we have X(8Kx,...,K d ) z> Y(8Ki,(K 2 -e),...,(K d -e)) * 0 . Then making

8 ^ 1

and keeping in mind the fact that X(Ki,.. .,1^) is a finite discrete set, we conclude that

X(Ki,...,Kd) * 0 .

1 < \N(a)\ <

Thus for

a e

X ( K i , . . . , K d ) , we see that

372

Answers to selected exercises

5. For any proper subset S of {1,2,...,r+t}, interchanging the order of infinite places, we may assume that {l,...,r+t-l} z> S. Consider the map 1 1 / : O* -> R^ " given by Z(e) = (log I e (i) | 5 ), where 8 = 1 or 2 according as (i) is real or complex. Then as shown by Dirichlet's theorem and its proof, l(cf) is a discrete submodule of RT+t~l and Rr+t"V/(C^) is compact. Thus for a sufficiently large M, any vector v e RT+t~l can be brought into the domain D = {(xi)e R r+t-l

< M for all i}

by translation by an element of /(0*). We take v e R^1"1 so that vi = N for some N > 0 and for i e S and Vi = -M for i e So-S. Then we pick 6 e Ox so that v-/(e) e D. Then -M < N-log I e (i) | 8 < M for i e S and -M < -M-log I e ( i ) I 5 < M, so log I e ( i ) I 6 > N-M for all i e S and -2M < log | e (i) | 5 < 0 for i e So-S. Thus by making N sufficiently large, we see that log I e (i) I 5 > 0 for all i e S and log I e (i) I 5 < 0 for i e So-S and Ilt\llog | e (i) I 5 > 0. Thus I e (i) I > 1 for all i e S and I e (i) | < 1 for all i e S n S 0 and n ^ ' 1 1 e (i) l 6 > 1. Since iT^i I e (i) I 8 = \N(e)\ = 1 , we see that | e (r+t) | < 1. This shows the desired assertion. §2.1. 3. (a) Note that if z stays in a compact set inside C-Z, we can find a positive integer M such that |z + n

ST-i 1 ^

+

i l

5M

< Mn" 2

+

^2)-

for all n > 0,

and hence

z-n| Thus

XL ( ^

+ ^ ) converges absolutely

and locally uniformly on C-Z. Similarly, if z stays in a compact set in C-Z, we can find e > 0 such that | ^ |

> en" 1 . Thus J^=1

| ^ |

> e ^ = 1 n"1.

Since Xr=i n"X

>

of y °°

because in an absolutely convergent series, any partial series is

M+1~+T^X

"-* +o° as m -^ +©<>, we know the divergence

also absolutely convergent. (b) We shall first show the absolute convergence of Xk=iX n =i n" 2k z 2k-1 when

| z | < 1 . We see that X ^ S ^

I n " 2 ^ " 1 1 < ^ = 1 k(2k)| Iz1 2 k l - Note

that

I £(2k) I < 1 +Jx"2kdx = 1+ Thus

= l4-(2k-l)"1 < 2.

Answers to selected exercises

373

This shows the absolute convergence of Xk=iXr=i n'^z21^"1. Since the limit value of the absolute convergent series does not depend on the order of summation, we know that 2-rk=lAi=l which was to be shown. 4. Write F(x) for - ^ - . 1+e

n

z

" 2^n=l2.k=l

n

Z

'

Then F ( - x ) = ^ - ^ - = -^— = l-F(x), and we see 1+e 1+e ) m F ( x ) l x = 0 for m> 0.

(

d \m

I

•^j F(x) I m = 2n, we know formally, by the functional equation, that

= 0. Writing

2r(2n+l)cos(n7T+(7c/2)) Note that £(s) is finite at s = 2n+l but cos(nrc+(7i;/2)) = 0. Thus £(-2n) to be 0 for the validity of the above equation. Thus we know that

nas

C(-m) = (^-)mF(x) | _0 = 0 for even positive integer m. 5. Looking at the expression

we know that (t^-) ^(t) has the expression

for a polynomial P m (t) with integer coefficients. induction. Then of course, we have P

m(t)

I

This can be proven by

-m-lz

This shows the result. 6. (b) Suppose p appears in the denominator of C(l-k). Then by Exercise 5, for any integer a > 1, a k (a k -l)£(l-k) e Z. Thus if a is prime to p, a k -l must be divisible by p, that is, ak = 1 mod p. We choose a so that a mod p gives the generator of the cyclic group (Z/pZ) x . Then ak = 1 mod p if and only if k = 0 m o d p - l . §2.2. 3. (b) First of all, when s = 1, Jp(8)G(y)dy = Ja D(e) G(y)dy = (2!Ci)Resyas0(G(y)) = 2TCI. On the other hand, we know that

374

Answers to selected exercises

Res s= i(e 27tis -1)- 1 = (2JU)"1. Thus by (e 2 ™-l)r(s)C(s) = JP(e)G(y)ys-1dy, we have Res s= iC(s) = 1. 4. (a) By the functional equation, we get

Since £(2n) > 0, we know the signature of B n . 6. We give the argument for the integral over Q/(m) since we can treat Qr(m) in the same way. For z on Q/(m), z = -(2m+l)+yL Thus I e"z I = e 2m+1 and I l-e"z | > I e"z | -1 = e 2m+1 -l. Therefore i

i

I I

e2m+1

^ ^

2

on

if m > 0

Q/(m)

-

This gives an estimate: |J QKm) G(z)z s - 1 dz| = \\(^li)n K2xn+i),

|

G(-(2m+l)+yO(-(2m+l)+yOs-1dy|

|o.ld

<2f2m+1* J-(2m+l)7t

2 2 (a 1)/2 dy (( 2m + l) + y ) -

dy J

<47t(2m+l) a - ^ 0 as m -> +«> (if a = Re(s) < 0). §2.3. 4. By Corollary 2, LQ-n.x"1) = 0 if %(-l) = 1 and L(l-n,xA) erwise. Then by the functional equation G(x)(27i/N)sL(l-s,%-1) . f

is finite oth-

Us y) = S 2V:ir(s)sin(7is/2) if %(-l) = -l, r(s)sin(?rs/2) is finite at s = l and hence L(s,%) is finite at s = 1 and if %(-l) = 1, the simple zero of r(s)cos(7i;s/2) is canceled out by the zero of L(l-s,%"!) at s = 1 and hence L(s,%) is finite at s = 1. 5. Since the argument is essentially the same in the two cases where %(-l) = ±1, we only treat the case of %(-l) = 1. Then by the functional equation, we see that =

2r(s)cos(7is/2)L(s,%) (2TC/N)SL(1-S,%-1) n

'

Expanding L(s,%) = Zn>mam(s-2) with am * 0, we see that

Answers to selected exercises

1 = L(s, X ) = Z n , m a m ( s - 2 ) n

375

(27C/N)(-I) 2m |a m | r

and

Applying the functional equation for the Riemann zeta function (i.e. replacing G(x) by 1 and % by the trivial character in the above formula), we know that r ( i ) = VTC because r(y) > 0 and hence G(x)G(x"1) = N = %(-l)N. When X is real valued, X"1 = %> am e R and by the above formula, we see that G(x) = (-l) m VN.

Thus if the order of zero at y *s e v e n (i n particular if

L(^,X) ^ 0), we know that

G(x) = VN = ^/%(-l)N.

§2.4. 1. (a) For a = I a I e10 e H' (I 9 I < y) and s = c+ix (a,x G R), we see that I a"s | 2 = a"s(a)"r = I a | " 2a e 20T . Thus if s stays in a compact subset of C, then | a"s | < M | a | "2Re^s) for a positive constant M independent of a. When A is not real, then I L*(n+x)" s | < Mllf =1 1 Re(aii(ni+xi)+-"+ari(nr+xr))+Im(aii(ni+xi)+'--+ari(nr+xr)) | "ai '

ai

if Gi>0.

Thus we may assume that A is real by replacing A by Re(A). When A is a real matrix, let M > 0 be the minimum of all entries of A. Then I L*(n+xy s | = n" 1 (aii(ni+xi)+...+a ri (n r +x r ))" Oi < n^iCMCm+xO+^+M^+x,))- 0 1 < M' Tr(o >n? 1 (ni+...+n r ))" Oi if G{ > 0 and not all ni are zero (if <J{ is negative, then replacing M by the maximum of the entries of A, we can deduce a similar inequality). Thus we may assume that all the entries of A are equal to 1, xi = 1 for all i and %i = 1 for all i. Then

I C(s,A,x,x) I < S n G Z + r(n 1 + -+n r )- T r ( a ) . We see easily that #{(ni,..., nr) e Z+r | ni+---+n r = k} < Ck r-1 for a positive constant C and hence I n e Z + r ( n 1 + - + n r ) - T r ( o ) < c X ^ k ' - 1 - 7 ^ = CC(Tr(a)-r+l). Since the Riemann zeta function ^(s) converges if Re(s) > 1, ^(s,A,x,x) converges if Re(Tr(s)) > r and Re(si) > 0 for all i. (b) It is sufficient to show that

376

Answers to selected exercises

B(r,k) = #{(m,..., nr) e Z + r | m+---+n r = k} > ck1""1 for a positive constant c, because then cC(Tr(a)-r+l) < X n e Z + r (ni+--+n r )- T r ^. We see easily that B(2,k) = k+1 and hence we can take 1 as c in this case. Supposing that B(r-l,k) > ckr"2 for some c > 0, we prove the assertion for r. We see easily that

B(r,k) = IJLoBCr-lj) > cIJLof2 > cj*xr"2dx = C -£p which shows the desired assertion. 3.

We have

(e 47C/s -l)(e 27C/s -l)r(s) 2 C((s,s),A,x,(l,l)) where _ exp(-xiu(a+bt)) exp(-x2u(c+dt)) " l-exp(-u(a+bt)) x l-exp(-u(c+dt)) _ exp(-xiu(at+b)) exp(-x2u(ct+d)) - 1 . e x p (. l l ( a t + b )) x i-exp(-u(ct+d)) Since

and Km (e 4it "-l)(e 2 *»-l)r(s) 2 =

«»inv

2<2%i)

we see that = the coefficient of u 2 ^ 1 ^ 1 of

= the coefficient of t11"1 of

(

nk+i

- k + j = 2n t k>0, j^O^"1^

X

B

k(xi)Bi(x 2 ) k!j!

i^a+P=n-1, a>0, P>ol a I (3

where we have used the binomial formula (X + Y) s = i ;

\

((n1)!) 2

Answers to selected exercises fll for

if n = 0,

( J ) = js(s-l) (s-n+1) I n!

377 1

if

n

>

0

(in particular, ('*) = (-l)n)

Therefore, the final formula is

.D) = 2-VDl

§2.5. 2. Let a and w be two ideals in O, and first show that there is a e O such p/1 (ei > 0). Put that aO = aS and £ is prime to m. Write w = p i e i c = ap\p2 pi and c\ - c/p{. Then q Z) c, and the two ideals are distinct. Thus we can find (Xi e q-c. Then oci is in apj for j ^ i but is not contained in ap[. If a is in api, then oci belongs to ap\ because all other Oj (j ^ i) are inside ap19 a contradiction. Thus a = oci+---+ar is not contained in ap\ for all i. Since a e a, we can write aO= aS and B is prime to w. Now, fixing a class C of ideals, we show that there exists an ideal de C such that d is prime to a given ideal nu In fact, ^ can be taken inside O. First take an integral ideal c in the class C and write c = CQC SO that the prime factors of CQ consists of those of m and c is prime to m. Take arbitrary O^yeco and decompose yO = aco. By the above argument, we can find a e O such that aO= aB and 5 is prime to nu Then a c = aBc J£ — = t£ c.

and the integral ideal (m,n)

(moc(z)+n) = m ^ ^ + n = (cz+d)"1((ma+nc)z+(mb+nd)). Thus Ek,N(cc(z),s;a,b) = I ( m , n ) e (a)b)+Z 2 (ma(z)+n)"k | (ma(z)+n) I "2s

378

Answers to selected exercises

= £(m,n)e((a,b)+z2)(X (mz+n)* | (mz+n) | -2s(cz+d)k | cz+d | 2 s = S(m,n)G((a,b)+z2) (mz+n)-k | (mz+n) | "2s(cz+d)k | cz+d | 2 s = Ek)N(z,s;a,b)(cz+d)k I cz+d I 2s for a e T(N). Thus Ek,N(z»s;a,b) is a modular form on F(N) of weight (k,s). §2.6. 1. We have a non-trivial unit e such that 0 < e < 1 < e a . % : Km) -> C x is an ideal character such that x((a)) = a k a a m , then k om . a k a a m = % ( ( a ) ) = % ( ( 8 a ) ) = (ea) (ea)

Then if

Thus e k e o m = 1. By taking logarithms of both sides, we see that klog(e) + m/<9g(e°) = 0, which yields klog(e) = mlog(e) because ee a = 1. Since logiz) * 0, we know that k = m. 3. Writing h for the class number of Q(V~P) 1

an(

i %( a )

=

( t ) ' we have

=-hp. Since p = 3 mod 4, %(-l) = -1, %(p-a) = -%(a) and

= liP=";1)/2(Z(a)a+X(p-a)(p-a)} = 2^ 1 ) / 2 X(a)-pll P = -; 1 ) / 2 X(a) = -hp. Similarly

Thus combining the above two formula, we have where A (resp. B) is the number of quadratic residues (resp. non-residues) modulo p between 1 and ?. By the quadratic reciprocity law, %(2) = 1 or -1 according as p = 7 or 3 mod 8. This shows (1) and (2). §3.5. 1.

We know from

ft—\ dn =

dt

J

. This shows

2. By (2b), we see that

(la)

that

J ( ^ydjiicp = (3nO(p) | T = o .

We write

Answers to selected exercises

379

« W ( t - D = Jtxd(q>*\|/)(x) = JJtx+ydcp(x)dV(y) = JJt x+yd9(x)d V (y) = Jtxd(p(x)JtydV(y) =* 9 (t-l 3. We know that Zp[T]] is the completion of R under the /rc-adic topology for m = (p,T) (T = t-1). Thus R = Z p[[T]]DQp(t), where the intersection is taken in the quotient field of Z p[[T]]. Since Zp[[T]] = ^ea*(Z p;Z p), the operator [<|>] preserves Zp[[T]] for any locally constant function $ on Z p having values in Z p , because the operation [$] : d-jLL h-> tyd\i preserves the measure space. On the other hand, by definition [0] preserves Qp(t). Thus [<|>] preserves the intersection R. §3.6. l.(ii). By definition, Rq, = R/(p(Ker(e'))R. Thus we have a sequence of groups: 0 -» Hom A -aig(R(p,S)--^HomA. a i g (R,S)

I!

II

9

* )HomA-aig(R'»S)

, II

n 9 0 — > Ker( G(S) * > G(S), where K : R -» R 9 is the projection map. The injectivity of TE* follows from the surjectivity of n. An algebra homomorphism <>| : R —> S is in Ker((p*) if and only if the following diagram is commutative:

R1

^ U

1

ie A

R

1$ ->

S.

Thus (|) e Ker((p*) <=> cp(Ker(e')) c Ker(<|>) <=> <>| induces such that 71*$' = 0. This shows the exactness at G(S).

(()' : Rq, -> S

§8.1. 1. Since the adele ring FA is the restricted direct product with respect to CV, for v outside m, we can consider an adele a in d(m) whose v component is 1 for almost all v and is G3V-1 for some finite number of places v outside m, where 05v is a prime element of Ov. Then 1+a in l+d(m) has C5V for some finite number of places. On the other hand, the v-component of any idele in d(m)x is a unit in CV for all finite places. Thus l+a£ d(m)x. §8.2. 1. If x n e FA X converges to x, then for sufficiently large n, (XnX'^f falls in the neighborhood U(m). Thus A,((xn)f) = X(xf). On the other hand, A,((xn)oo) = (xjeo^ is a polynomial function on the vector space F~ = fly realR X U, complexC (1.1.5b),

380

Answers to selected exercises

which is obviously continuous. Thus lim X,((xn)«,) = ^(Xoo) and hence lim X(xn) n—>©o

n—>«»

= A,(x). That is, X is continuous. 3. (i) Since Cl(m) is a finite group, if one writes h for its order, for any ideal a prime to m, a*1 = aO for a e P(w). Thus X*(a)h = X*(aO) = a^, which is contained in the Galois closure O of F over Q. Thus X(d) is always algebraic. If {«i}i=i h is a representative of classes modulo m, then any a can be written as ao{ for one of the o^'s. In particular, X(a) = a^X(oi) s K = d>(X(ai) | i=l,..., h). The field K is generated over O by finitely many algebraic numbers and hence is a number field. Since X(x) = X*(xO) for x e FA(W), we know that X(x) is contained in a number field K independent of x e FA(W). (ii) This follows from (1.3.4b). (iii) (A. Weil) Writing x = auax^ e F A X for a e F x , UG U(OT), a e FA(mp) and Xoo e Foo+, we define ^P(X) = ?L*(aO)up~^, where up is the projection of u to n v | p F v x . If there are two expressions ocuaxoo = cc'u'a'x'oo, then a p u p = oc'pu'p because (axoo)p = (a'x'oo)p = 1. On the other hand, oc'^a = u'u^a'a^x'ooXoo"1 e U(wp)FA(/rtp)Foo+nFx = P(wp). Thus a'O = a'^aaO and we see that

Thus the character Xp is well defined. The continuity can be verified as in Exercise 1 replacing Foo by F p in the argument there. §8.3. 1. Suppose that a n : K —> T is a sequence of continuous homomorphisms with a n (x) = \j/(ynx) for yn e K and that a n converges locally uniformly to a. We want to show that yn converges to y e K given by oc(x) = \|/(yx) for all x E K. Since ccn converges to a, for any given m > 0, ocn-a has values in Ii on %'mOv for sufficiently large n and hence by Lemma 1, a n = a on 7i~m0v. If rcr0v is the different of \|/, then a n = a on 7i"mOv means that yn-y e 7Cm"rOv and hence making m large, we know that yn converges to y in K. Conversely, if yn converges to y in K, then reversing the above argument, we know that c^ converges to a. Thus the isomorphism of Proposition 1 is also the topological isomorphism. §8.5. 1. (ii) Note that a+py1 = a(l+a'1G31Ov) and thus by definition,

Answers to selected exercises

a

381 ) if | a | v >

Therefore, for any locally constant function (j) factoring through (Ov/pv1)x,

Any locally constant function on CVX is of the above form for sufficiently large i and therefore the above formula is true for any locally constant functions on CVXThe formula holds even on F v x because F v x = Ui C51OVX. (iii). By the above formula, | x | v"1dji(x) gives a multiplicative Haar measure. Therefore J F (|)(ax) I x | y'MjiCx) = JF (|)(x) | x | v^djuiCx) for any locally constant function (J). Any locally constant function <|) can be written (|)(x) = f(x) | x | v for another locally constant function f, and the correspondence <>| h-» f is bijective on the space of locally constant functions. Replacing <|) by f(x)|x| v in the above formula, we see that

I a I vJFvf(ax)dja(x) = JFyf(ax) | a | v dn(x) = JFvf(x)d^i(x), which proves the assertion. §8.6. 1. Using the fact that F A X /F X = F ^ / F x x R > 0 via x h-> (xf(

*~.Q]), | x | A)

(where R>o = {x e R | x > 0}), we can write ^(x) = Xi(x)X2(x) for characters Xi of F ( A } /F X and X2 of R >0 . Since F^/F* is compact (Theorem 1.1), ?ii must have values in T, which is the unique maximal compact subgroup of C x (in fact C x = TxR>o). Any continuous linear map of R into C is given by th->cct with a e C . By identifying R with R>o via "exp", any continuous multiplicative map of R>o into C x is given by xf-» xs with s e C. Thus

I X(x) I = I A,2(x) I = I x I As for some s e C . 2. Let % be as in the exercise. We see in the same manner as above that I %(x) I = I x I s for some s e C . Thus by taking %/1 % I instead of %, we may assume that % has values in T. We already know that Homcont(R>o,T) = R (assigning a e R to the character x h-> x10t) and Homcont(T,T) = Z (assigning n e Z to the character x h-> x n ). Since C x = TxR>o via x h-> (x/1 x |, | x | ) , we thus see from this that % has the desired form.

382

Answers to selected exercises

3. For any standard function Of on FA£, we can find a rational number N ^ 0 and real number M such that | Of(x) I < M | ^Ff(Nx) | for the characteristic function Wf of Zf. Thus we may assume that O = OfOoo and that Of is the characteristic function of Zf. We may also assume that Ooo(x) = xkexp(-7tx2) for 0 < k E Z. Then for £ e Q, O(£) * 0 if and only if £ e Z. This shows that 5^GQ ^C^) = £neZ nkexp(-7cn2), whose convergence is obvious from the convergence of geometric series Z°° exp(-7tn) and its derivatives £°° nkexp(-7tn). §9.1. 2. Since % : Cl(m) = A F x /F x U(m)F oo+ -* T, %v(Owx) = 1 if v is outside m and %(FX) = 1. If e e O*, then E G OVX for all finite v. Thus

Therefore

0 d

IIv | wZv(e) = TT^TT; , then d e O x . Thus

hand, j([j ^x o . ) i) k =

for all e e O x . X*

In particular, if • On the other

0 d

. Hence we have for

B..

Index

Adele 239 ring 239 adelic Fourier expansion 276, 277 adelic modular form 273 Adjoint formula (Petersson inner product) 315 Algebraic Petersson inner product 222 Algebraicity algebraicity theorem of standard L-functions of GL(2) 186 algebraicity theorem for Rankin products 324 Shimura's algebraicity theorem 310 Approximation theorem 274 strong approximation theorem 161 Arithmetic point 220 Automorphic property 126 Banach A-module77 Bernoulli number 37, 38 polynomial 42 generalized Bernoulli number 43 Bialgebra 90 Borel-Serre compactification 187 Boundary exact sequence 112, 352, 363 Bounded p-adic measure 78, 120 Class group 7 class number 7 class number formula 63 Coboundary 347 Cocycle 347

Cohomology group 345 compcatly supported sheaf cohomology group 360 Cech cohomology group 353 de Rham cohomology 359 group cohomology 345 parabolic cohomology group 347, 351 singular cohomology 345 Common eigenform 146 Commutative group scheme 91 Congruence subgroup 125 Conjugate 137 Constant sheaf 353 Convolution product 199, 327 Coset decomposition 139 Cup product 170, 183 Cusp 166 equivalence classes of cusps 148 cusp form 126 cuspidal condition 278 irregular cusp 166 regular cusp 166 Cech cohomology group 353 Decomposition group 24 Dedekind domain 11 zeta function 54 Derivation 92 Differential operator 92, 310 invariant differential operator 93 Dimension formula 160, 164, 166 Dirichlet -Hasse (a theorem of) 244 character 89

384

Index

unit theorem 15 class number formula 66 residue formula 65 Discriminant 10, 12 Discriminant function 131 Distributions 119 distribution relation 119 Dominated convergence theorem 36, 50 Dual group 250 Duality (a theorem of) 142, 150, 218 Eichler-Shimura isomorphism 160 (a theorem of) 168 Eisenstein series 54, 58, 125 (functional equation of) 292 Euler product 41 Euler's method 25 Exact 1 Extension functor 3 Extension module 1 Flabby sheaves 360 Formal completion 96 Formal group 96 multiplicative group 89 Fourier expansion 125, 128 adelic Fourier expansion 276, 277 Fourier expansion of f |T(n) 141 Fourier transform 112, 256 Fractional ideal 5 Free resolution 345 Frobenius conjugacy classe 24 element 23 Functional equation 26, 38, 309 of Eisenstein series 292 of Hecke L-function 261 of Rankin products 306 Gauss sum 45 Geometric interpretation 181 Group decomposition group 24 functor 94

scheme 92 formal group 96 idele group 241 inertia group 23 Haar measure 254 Hecke (a theorem of) 70, 72 algebra 141, 181 character 54, 63 L-function 54 module 177 operator 110, 139 ring 176 universal ordinary Hecke algebra 218 Hilbert modular Eisenstein series 71 form 72 Hodge operator 175 theory 171 Holomorphic projection 314 Homogeneous chain complex 349 polynomials 107 Homological method 107 Homotopy equivalence 345 Hurwitz L-function 41 Ideal group 54 Idele 240, 241 group 241 Imaginary quadratic field 63 Inertia group 23 Interpolation series 73, 75 Invariant differential operator 93 Involution 144 Irreducibility 165 Irregular cusp 166 Iwasawa theory 106

Index

A-adic cusp form 196 Eisenstein series 198 form 196 forms of CM type 237 ordinary form 208 Leopoldt conjecture 103 L-function abelian L-function 25 Dirichlet L-function 40 modular L-function 125 Hurwitz L-function 41 Shintani L-function 47 Lipschitz-Sylvester theorem 33 Locally constant function 84, 99, 118 Locally constant sheaf 353 Long exact sequence 2 Mahler (a theorem of) 77 Manifold 345 Mellin transform 186 Minkowski's estimate 10 Mobius function 286 Modular form of weight k 125 symbol 120 nearly holomorphic 314 Norm 6, 7 Normalized absolute value 242 eigenform 187 Number field 5 Numerical 73 polynomial 75, 93 One variable interpolation 225 Open covering 353 Ordinary 189 A-adic form 208 part 202 projector 202 Orthogonality relation 45, 46

385

p-adic analytic function 88 exponential function 21 family of modular forms 195 integer ring 19 Dirichlet L-function 87, 124 Hecke L-function 105 logarithm function 21 measure 78 meromorphic function 88 residue formula 105 standard L-function of GL(2) 160, 189 Mellin transform 160 number 17 p-fraction part 239 p-ordinary 189 Pairing 169 Parabolic cohomology group 347, 351 Period 187 Petersson inner product 143 Place 18 Poisson summation formula 256 Polyhedral cone 68 Presheaf 352 Prime 8 factorization 11 Primitive 44, 317 form 317 modulo 45 Product formula 242 Pseudo-representation 232 Ramanujan's A -function 131, 234 Rankin product 154 Rapidly decreasing 297 Rational structure 131 Rationality of the Dirichlet L-values 138 Ray class group 54, 245 Real quadratic field 54 Regular cusp 166

386

Index

Relatively prime 8 Representation pseudo-representation 232 residual representation 229 Represented 92 Residue formula 228, 271, 305 Restricted direct product 240 Riemann surface 345 zeta- function 25 Semi-group 175 Semi-simplicity 218 Shapiro's lemma 182 Sheaf 352 compactly supported sheaf cohomology group 360 cohomology 345 cohomology group 360 constant sheaf 353 generated by 356 locally constant sheaf 353 presheaf 352 Shimura's algebraicity theorem 310 Shintani L-function 47 Siegel-Klingen (a theorem of) 57 Siegel's estimate 10 Simplicial complex 348 cone 67 Singular cohomology 345 Slowly increasing 297 Smooth manifold 352 Space of modular form 126

Standard flabby resolution 360 additive character 249 function 261 L-function 157,186 L-functions of GL(2) 186 R[F]-free resolution 346 Strict ray class group 10 Strong approximation theorem 161 Subdivision operator 362 Symmetric n-fold tensor 169 Teichmiiller character 87 Tensor product 3 Theta series 235 Three variable interpolation 332 Topology of uniform convergence 250 Torsion functor 3 Totally real 4 Trace 6 map 111 operator 182 Transformation equation 137 Triangulation 362 Two variable interpolation 227 Universal ordinary Hecke algebra 218 Unramified 228 Upper half complex plane 58, 125 Weierstrass preparation theorem 209 Weil (a theorem of) 247