Galois Theory of Difference Equations

This last algebra can be identified with

M(p, 2 ~ ) . Consider f as 2,~-linear m a p between two free 2m-modules of rank p 2 By construction, the reduction modulo rr_2of f c o i n c i d e s w i t h the bijection Z/m| D -+ M(p, Z / m ) . Hence f itself is a bijection. By reducing f m o d u l o powers of rn one finds the other statements of 4. For the construction of the invertible element u of 2 ~ is suffices to prove the following assertion:

Let k > 1. Suppose that there is an invertible element uk in Zm[z] such that uk(z + p - 1 ) . . . u k ( z + 1)uk(z) is congruent to 7' modulo ,_nk. Then there is an element uk+l of the form uk(1 + Fkb), with F a 9enerator of the ideal 77~ and b E 2"~[z], such that Uk+l(Z + p - 1 ) . . . u k + l ( z + 1)Uk+l(Z) is congruent to T modulo m k + l .

,5.1. G E N E R A L I T I E S

55

A small calculation shows that the element b should satisfy b(z) + b(z + 1) +

... b(z + p - 1) is congruent modulo m with the expression T-

uk(z + p - 1 ) . . . u k ( z + 1)uk(z)

uk(z q- p -

1 ) . . . u k ( z + 1)uk(z)F k "

The element a is seen to lie in ,),~. The choice b = ~p - 1 * has the required prop-erty as is easily verified. 5. Since Z is a ring of Laurent polynomials over a field it follows that Z = f i n Z,~ where the product is taken over the set of all maximal ideals of Z. Then 5. follows at once from 4. |

P r o p o s i t i o n 5.2 The category of Z-modules of finite dimension over k(zP - z)

is equivalent to the category of the left D-modules which are of finite dimension over k(z). This isomorphism preserves tensor products. P r o o f . The category of Z-modules of finite dimension over k(zP - z) coincides with the category of Z-modules of finite dimension o v e r k ( z p - z ) . The same statement holds for I9 := Z | D. We use part. 5 of Proposition 5.1 and combine this with Morita's equivalence. This equivalence can be formulated as follows (See [54], Proposition 17, p.19):

Let A be any (unitary) ring. The categories of left A-modules and left M ( n , A)modules are equivalent. The equivalence is given by M ~ M ~, with the obvious left action of M ( n , A ) on M '~. This conchdes the construction of tile equivalence f : N ~-+ M between the two categories. A more explicit way is to define f ( N ) as k(z) Ok( . . . . ) N. Its structure as a k(z)[OP, O-P]-module is clear. The construction above leads to a compatible O-action on M. Finally we have to say something about tensor products. Let L be any field and let F M o d f [ r , r - q denote the category of the modules over LIT, T - I ] , which have finite dimension over L. This abelian category is given a "tensor structure" by defining the tensor product of two modules M and N as M | N with the operation of T given by the formula T(rn | n) = (Trn) | (Tn). This explains the tensor product for the first category of Proposition 5.2. The tensor product for difference modules M and N over k(z) is defined in a similar way, namely: The tensor product is M | N with the operation of 9 given by O(m G n) = (Orn) | (On). It is not difficult to show, using the description .T(N) = k(z) | . . . . ) N, that f respects tensor products. 1

CHAPTER. 5. A N E X C U R S I O N IN P O S I T I V E C H A R A C T E R I S T I C

56

5.3 For every maximal ideal m in Z and every n > 1 one defines the indecomposable left :D-module I(rn '~) to be Y ( Z / m ~) = ( Z / m " ) Gk( . . . . ) k(z). Every left ~D-module M of finite dimension over k(z) is isomorphic to a direct

Corollary

stlm

r?~, rl

The numbers e(m, n) >_ 0 are uniquely determined by M. P r o o f . T h e i n d e c o m p o s a b l e Z - m o d u l e s of finite dimension over k(z p - z) are the Z/mft. Hence Corollary 5.3 follows at OllCe from P r o p o s i t i o n 5.2. |

5 . 4 The p-curvature of a difference module M over k(z) is the z)-linear action o f alPp o n the module N with :F(N) = M.

Definition k ( z p --

By c o n s t r u c t i o n the k(z p - z)-linear dpp on N e x t e n d s to the k ( z ) - l i n e a r ~P on M = k(z) | N. We will also call qSp on M the p - c u r v a t u r e when no confusion occurs. T h e n a m e is copied from a similar s i t u a t i o n for differential e q u a t i o n s in c h a r a c t e r i s t i c p. Let M be a difference m o d u l e over k(z) which is r e p r e s e n t e d by an e q u a t i o n in m a t r i x form y ( z + 1) = A y(z), where A = A(z) is an invertible m a t r i x with coefficients in k(z). T h e n the p - c u r v a t u r e (as a k(z)linear m a p on M ) has the m a t r i x A(z + p - 1 ) . . . A ( z + 1)A(z).

5.2

Modules

over

I([T,T -1]

For the m o m e n t K is any field. T h e modules M over K [ T , T -1] are s u p p o s e d to be finite d i m e n s i o n a l vector spaces over K . In other words, a m o d u l e is the s a m e t h i n g as a vector space M over K of finite d i m e n s i o n t o g e t h e r with an invertible linear m a p (i.e. the action of T on M ) . T h e c a t e g o r y of all those m o d u l e s is d e n o t e d by FModK[T,T-1]. It is in an obvious way an a b e l i a n category. T h e s t r u c t u r e of the tensor c a t e g o r y is defined in the p r o o f of P r o p o s i t i o n 5.2. T h e functor w : FModK[T,T-,] --4 VectK, where VectK denotes the c a t e g o r y of finite d i m e n s i o n a l vector spaces over K , is the forgetful functor w ( M ) = M (i.e. one forgets on M the action of T). One can verify t h a t w is a fibre functor. T h i s makes FModK[T,T-~] into a n e u t r a l T a n n a k i a n category. A s s o c i a t e d to this is an affine g r o u p scheme G over K . T h i s affine group scheme represents the functor R ~-~ Aut| defined on K - a l g e b r a s . For a fixed o b j e c t M E FModK[T,T-1] one call consider the tensor s u b c a t egory { { M } } of FModK[T,T-1] g e n e r a t e d by M . T h i s is the full s u b c a t e g o r y whose o b j e c t s are the s u b q u o t i e n t s of the tensor p r o d u c t s

5.3, DIFFERENCE GALOIS GROUPS

57

M| M|174 | |174 (as usual M* denotes the dual of M). The restriction of w to {{M}} is again a fibre functor. The affine linear group associated to this {{M}} is denoted by GM. It is an exercise to show that GM is the smallest linear algebraic subgroup of Gl(M) which contains the action of T on M. We note that for an algebraically closed field K of characteristic 0, this group GM is a direct product of a torus, a finite cyclic group and possibly a Ga. The Ga is present precisely when T on M is not semi-simple. If the characteristic of K is p > 0 then (TM is a direct product of a finite cyclic group and a torus. Indeed, suppose again that K is algebraically closed and let T = T,~T~ be the decomposition of the action of T on M into a semisimple and a unipotent part. The group (TM is generated as an algebraic group by T~s and T~. The group generated by T,, is easily seen to be a product of a torus and a cyclic group with order prime to p. The group generated by T~ is finite of order p~ where n is minimal such that the nilpotent matrix T~ - 1 satisfies (T~, - 1)v~ = 0.

5.3

D i f f e r e n c e Galois g r o u p s

For difference equations over the field k(z) one would like to have a suitable theory of Picard-Vessiot extensions. The difference Galois group of an equation would then be the group of automorphisms of the Picard-Vessiot ring of the equation. It is not excluded that such a Picard-Vessiot theory exists, in the sequel however we will use the theory of Tannakian categories tbr the definition and study of the difference Galois group. The main idea is to compare difference modules over k(z) with modules over Z = K[T, T-l], where K = k(z p - z). Let M be a difference module over k(z) and let N be the Z-module with .T(N) : M. The category of difference modules {{M}} is defined in a way similar to Chapter 1.4. The functor .P induces an equivalence {{N}} -+ {{M}} of tensor categories. Hence {{M}} is also a neutral Tannakian category with fibre functor

The difference Galois group of M is defined as the linear algebraic group over K associated to this fibre functor. This group is of course isomorphic to the group GN of N defined in Section 5.2. Thus we find the following properties:

C H A P T E R 5. A N E X C U R S I O N IN P O S I T I V E C H A R A C T E R I S T I C

58

1. The difference Galois gTvup of a difference equation over k(z) is the algebraic group over k(z p - z) generated by the p-curvature r 2. The difference Galois group is a direct product of a finite cyclic group and a torus.

5.4

C o m p a r i n g characteristic 0 and p

We s t a r t by considering the following e x a m p l e of an order one equation:

y(z + 1) - z- +- 1 / 2 y ( z ) over ~_~,z,. O( Z

T h e only algebraic solution of this e q u a t i o n is 0. However, for every p r i m e p > 2 the p - c u r v a t u r e is

z+p-l+l/2 z+p-1

z+1+1/2 z+l

zr + 1 / 2

m

].

z

T h i s is clearly a counter e x a m p l e to the naive t r a n s l a t i o n of G r o t h e n d i e c k ' s c o n j e c t u r e for difference equations. How to explain this? A s o m e w h a t trivial e x p l a n a t i o n is t h a t the r e d u c t i o n of a r a t i o n a l n u m b e r m o d u l o a p r i m e is an integer m o d u l o a prime. T h i s leads to the following result.

L e m m a 5.5 Consider the equation y(z + 1) = ay(z) with a C q ( z ) * . following statements are equivalent:

The

1. For almost all p the p-curvature is 1. 2. a has the form b(z+A) b(z) where b E Q ( z ) and ~ E Q 3. a(oo) = 1 and for every algebraic number (~ the restriction of the divisor of a to any Q-orbit <~+ Q C Q has degree O. P r o o f . C l e a r l y 2. implies 1. and 2. implies 3. To see t h a t 3. implies 2. note t h a t there exists an integer q such t h a t the r e s t r i c t i o n of the divisor of a to a Q - o r b i t a c t u a l l y lies in a set of the form c~ + ~-Z. Let b(z) = a(~). q T h e r e s t r i c t i o n of the divisor of B to a Q - o r b i t is now a c t u a l l y a Z - o b i t . ' ~ e can therefore a p p l y P r o p o s i t i o n 2.1 to conclude t h a t b(z) - g(z+l) T h e r e f o r e --

9 ~ ) - -

'

a(z) = g(q~+l) 9(qz) " Dividing the n u m e r a t o r and d e n o m i n a t o r of this q u o t i e n t by a s u i t a b l e power of q, we have t h a t a(z) = h(z-j-1/q) for a s u i t a b l e h. h(z) Now s u p p o s e t h a t 1. holds. T h e n a c a n n o t have a pole or zero at oo and the value a(oo) E Q* has the p r o p e r t y t h a t for a l m o s t all p, one has a(oo) p - 1

5.4. COMPARING C H A R A C T E R I S T I C 0 AND P

59

mod p. This implies that a(oo) = 1. Write a = a l . . . a s , where the support of the divisor of each ai lies in the union of {oc} with the Q-orbit of an algebraic number ai. The c~i are supposed to be distinct modulo Q. Each term ai has the form (z - o~i)n'bi, where ni E Z and where bi satisfies condition 3. The primes p that we consider are now prime ideals :fi 0 in the ring of integers of the field Q(c~l . . . . ,(*~). For almost all p the reductions modulo p of the c~i are distinct modulo Fp. The p-curvature for almost all primes is (Z p

--

(~I)P--1) n~

.

.

.

(Z p

--

(~s)P-1)

n~

z 1 m o d p.

There is no cancellation between the various factors. Hence all rzi = 0 and a satisfies 3. I Another interesting example is z2 -2

y(z + l ) -

z2

y(z)

WTe claim that for p > 2 the p-curvature is 0 if and only 2 is a square modulo p. Using the formula 1-Ii=0 ..... p _ l ( z + i - a ) - z P - ( a ) P - l z mod p, one finds that the p-curvature of the expression above is (zP-2cP-1)/~z)~ (. . . . )~ mod p. The p-curvature is 1 if and only if 2 ( p - l ) / 2 _= 1 mod p. This proves the statement. Finally, we give a general example where p-curvature 1 occurs for ah-nost all p. Suppose that the equation 9(z + 1) = A 9(z) with A E Gl(n,Q(z)) has a formal fundamental matrix F(z) E Gl(n,Q((z-1)) ). Suppose moreover that a reduction modulo p of F exists for almost all p. This is the case when there are only finitely many primes in the denominators of the coefficients of the entries of F. Then, for almost all primes p, one has the equality A(z + p - 1)...A(z + 1)A(z) = F ( z + p)F(z) - t - 1 mod p. These examples give the impression that the p-curvatures of an equation over Q(z) contain arithmetical information about the fundamental solution of the equation.

Chapter 6 Difference

6.1

modules

over

Classification of difference modules

7)

o v e r 7)

We recall that 7) is the algebraic closure of C((t)) where t = z -1 and r = r + r A one dimensional difference module over P is given as 4PeS = fef with f E 7)*. Let us call this module g(f). Clearly $ ( f l ) and g(/2) are isomorphic if and only if flf21 has the form r for some a E 7)*. Write a as a 0 t ~ ( l + ~ a . t ~) with a0 E C* A E Q a . ~ C # > O, # E 2 Z for some m -> 1 (depending on a). A calculation of -r- 7 consisting of all elements @

U:={(I+At+Eb~t~)[AEQ, p>l

yields that the subgroup U of P*

is equal to bt, E C, # E

2z for s o m e m > 771

1}

For the group P*/U one can take various subgroups g of 7)* as set of representatives. We will take g as follows: = {t " x c ( 1 W t ) a ~ 1 6 2

cEC*, q=

E

a.z"}

0<#
where # E Q, a~ E C and where the sum ~0
LoQ=c. Any 1-dimensional difference module over 7) is isomorphic to a unique g(g) with g E 6- Furthermore there is a natural isomorphism g(gl)| --+ g(glg2), given by em | eg~ ~ eg~g:. We note that our choice for g is inspired by the interpretation of the formal solution of the difference equation as (the inverse of) the multivalued function rXc'z~~ A difference module M is called unipotent if there exists a sequence of submodules 0 = M0 C M1 C ... C Mr = M such that every quotient Mi/Mi-1

6.1.

61

CLASSIFICATION OF DIFFERENCE MODULES OVER P

is isomorphic to the unit object g(1). The subcategory of unipotent difference modules over P is closed with respect to direct sums, duals, subquotients and tensor products. A unipotent module is called decomposable if it is a direct sum of two other modules. The following lemma gives the structure of the unipotent modules. L e m m a 6.1 1. The cokernel of the map ~ - 1 : 7) --+ P has dimension 1. A basis of this space can be represented by t = z -1. 2. For every n >_ 2 there exists a unique indecomposable unipotent module M,~ of length n. The matrix of 9 with respect to a suitable basis of M~ has the f o r m

10 0)

s

0

1

0

0

0

s

1

where s is any representative of the cokernel of r - 1. 3. The decomposition of a unipotent module M as a direct sum O~_>zM~ ~ is unique up to isomorphism. The integers aN >_ 0 are unique and determine M. 4. A n y unipotent difference equation has a f u n d a m e n t a l matrix with coefficients in the simple difference ring 7)[X], given by r = X + t.

Proof." The proof of 1. is an easy calculation. We prove 2. by induction on the length of M. A unipotent difference module of length 2 has a basis such that the matrix of q5 in this basis is us call this module M ( f ) . if and only if

f

1

"

Two elements fl, f2 E ~ define isomorphic modules

f l : c f2 ~- d(g) -- g with c E C* and g E ~)

Letting s be a representative of the cokernel of r 1, it follows that there are only two non isomorphic modules of length 2, namely M(0) and M ( s ) . This proves the case n = 2 of part 2. of the lemma. We will prove the case n = 3 and omit the full proof of 2. By induction an indecomposable unipotent module of length 1 0 0) s 1 0 with f E 3 has a basis for which the matrix of 9 has the form f s 1 1 0 0) "P. Let B = c 1 0 withgE7 ) andcEC. g 0 1

CHAPTER 6. DIFFERENCE MODULES OVER 7)

62

1 0 0) s 1 0 r is a new matrix for ~. A calculation shows f s 1 that for suitable c and 9 this new matrix is the matrix of the l e m m a for n = 3. In order to verify, that Ma as given in ~he l e m m a is indecomposable one has to Then B -1

check that the equation v =

(100)

s 1 0 s space over C. This is an easy exercise.

0 1

r

has a 1-dimensional solution

Part 3. of the l e m m a follows from the Krull-Remak-Schmidt Theorem. The first thing to prove for part 4. is two show t h a t 7)IX] is simple.

Let

f = X d + a a - l X d - l + . . . + a o with d > 0 be such that (f) C 7)[X] is r Then &(f) - f has degree less than d and must be 0. Comparing the coefficients o f X d-1 of r and f one finds the equation dt+r = ad_l. This equation has no solution in 7). We conclude that 7)IX] is simple. It suffices to show that M~ has a full set of solutions over 7)[X]. This amounts to solving

or

r

) -

Yl =

0,

0(?)2)

-

'0 2 =

~Yl,..

One easily finds t h a t vl is any constant, v: = Xvl, v3 = ~ X ~ + something in P , etc. This proves part 4. | There is a more efficient way to deal with unipotent difference modules. We will compare these modules with the differential modules over 7) which are nilpotent. Let d denote the differentiation on 7) given as z ~--~, i.e.,

a ( ~ a ~ t ~) = ~ a ~ ( - p ) t ~. Let 7)[~] denotes the skew polynomial ring given by the formula da = ad + d(a). A differential module over I) is a left 7)[@ module which is of finite dimension over 7). We write s for the unit object in the category of differential modules over 7). Thus g = 7)e with de = 0. A differential module M over 7) is called nilpotent if there exists a sequence of submodules 0 = M0 C M1 C ... C M~ = M such that every quotient Mi/AJi_l is isomorphic to the unit object g. It. is well known that every nilpotent differential module M has the form M = 7),~c W, where W is a vector space over C invariant under the action of d and such that the restriction of d to W is a nilpotent linear map. We note t h a t W is unique since it is the kernel of the operator d ~ on M for n _> the dimension of M over 7). In order to associate to a nilpotent differential

THE UNIVERSAL PICARD-VESSIOT

6.2.

R I N G OF 79

63

module M a unipotent difference module, we have to introduce some notion of convergence. The field P has a valuation given by IN[ = e x p ( - , X ) i f the expansion of f E 79* starts with the term t x and of course 101 = 0. The field 79 is not complete with respect to this valuation but all the subfields C ( ( t l ) ) are complete. A vector space N of finite dimension over 79 is given a norm by choosing a basis el,..., % and by defining II ~ a~e~l] = m a x lail. T h e topology induced by this n o r m does not depend on the chosen basis. Let M be a nilpotent differential module over 79. One defines an operator OM on M by the formula

n>0

One can verify that (tS) ~+1 = t n + l ( 6 - t I ) ( ( 5 - n q- 1)...6. This implies that the infinite expression for r converges for every m E M. Further one can verify that the difference module M defined by ~ ( m ) = eM(rn) is a unipotent difference module. On the other hand, a unipotent difference module M induces a nilpotent differential module by defining the action of 6 on M by the fornmla _(-i)'~+i _ ( ~ ,

6 = t-llog (~) := t -1 ' ~

- ~),,.

7~ n>

1

The procedure above reflects of course the property that the a u t o m o r p h i s m r of P is the exponential of the derivation ~ = - t ~"~7 d o n "P. The correspondence between unipotent difference modules and nilpotent differential modules is an equivalence of tensor categories as one easily sees. As is well known nilpotent modules have a full set of solutions in the dilferential ring 79[l], where ~ I := 7' l In view of this correspondence with differential equations we prefer to consider the simple difference ring 79[X] with r - X = log(1 + t). In the sequel we will write / for X and we note that l has as interpretation the multivalued function log z. T h e o r e m 6.2 Every difference module M has a unique decomposition @gE~Mg, such that M e = g(g) @. M(g) and M ( g ) is a unipotent difference module. Moreover for gt -7k g=, the vector space Hom(Mg~, M ~ ) is O. A proof of this statement follows easily from [21] and [47].

6.2

T h e universal Picard-Vessiot ring of P

Define the ring R = 79[{e(g)igco][I ] with generators {e(g)]ge~ a n d / a n d relations = e(gl)e(g~e), e(1) = 1. Thus R is the polynomial ring in the variable

e(glg2)

C H A P T E R 6. D I F F E R E N C E M O D U L E S O V E R P

64

l over the group algebra of the group ~. The action of r on /~ is given by r = ge(g) and r - l = log(1 + t) C C{t} C 7) . We will first show that R has only trivial r ideals. Let I be a nonzero ideal of R with r C I. Let f C I be a nonzero element such t h a t its degree d as a polynomial in l is minimal. If d > 0, then one sees t h a t r - f is a nonzero element with smaller degree. Hence J := I C~7)[e(g),g E g] is nonzero. Choose an element f = E i = I ...... aie(gi) # 0 in J with r minimal. If r = 1 then f is invertible and so I is the trivial ideal R. I f r > 1 then after multiplying f with a l l e ( g l ) -1 we may suppose that al = 1 and gl = 1. T h e n r - f = ~ i = 2 ...... (r - ai)e(gi) = 0 by the definition of r. The term ai E 7) is not zero and satisfies r = ai. This contradicts the construction of g and we have shown that R has only trivial r ideals. .

Next we will show that C is the ring of constants of R. Let f be a nonzero constant of R. The ideal (f) is r and from the above it follows that f is a unit of R. In particular, f has degree 0 as polynomial in l and f = ~ i = 1 ...... aie(gi) with r > 1 and all ai :/: 0. The equation r = f and the definition of g easily implies that f = ce(1) with c C C*. Every difference equation over 7) has a fundamental matrix with coefficients in R. Indeed, using the classification of Chapter 6.1, one finds that it suffices to consider the difference modules g(g) and the unipotent difference modules. For the first one e(g) is the fundamental matrix. The unipotent case has been done in L e m m a 6.1. Clearly R satisfies the minimality condition. This proves that R is indeed a universal Picard-Vessiot ring for 7). We will now show that any universal Picard-Vessiot ring for 7) is isomorphic to the R above. Let B be another universal Picard-Vessiot ring and let b(g) denote the invertible element of B satisfying r = gb(9 ). Let L be an element of B satisfying r = L +1o9 (1 + t). One can show that a normalization of the b(g) exists such t h a t b(1) = 1 and b(gl)b(g2) = b(glg2) for all gl ,g2 E g. Indeed, one considers the multiplicative subgroup H of t3" consisting of the elements of the form ab(g), with a E P* and g E G. The smjective h o m o m o r p h i s m H --+ g has a right-inverse since g is a divisible group. Let g ~-~ b(g) denote the right.inverse. Then clearly b(gx)b(g~) = b(glg2). The obvious surjective, r 7)-algebra h o m o m o r p h i s m F - R --+ B, e(g) ~+ b(g), l ~-+ L has kernel 0 since R has only trivial r ideals. This ends the proof. The following result is now an easy consequence. C o r o l l a r y 6.3 Let the difference field k be a difference subfield of 7) with field

of constants C. The functor w :Diff(k,r

~ Vectc

given by ~ ( M ) = k e r ( ~ - 1, R O k M) is an exact, faithful morphism of C-linear tensor categories.

6.5. THE FORMAL GALOIS GROUP

65

As a final remark, we note that the ring R has zero divisors. This is due to the fact that the group g has elements of finite order. Indeed, the roots of unity #o~ in C* C g is the torsion subgroup o f ~ . Put e .... := e(e~:~). The factorization of T m - 1 in linear polynomials yields zero divisors after substituting e,= for T.

6.3

Fields of c o n s t a n t s which are not algebraically closed

Let k C 7) denote a difference subfield such that the field of constants k0 of k is not algebraically closed. For convenience we will continue our discussion with the case k = ko(z) and C is the algebraic closure of k0. There is a natural action of Gal(C/ko) on 7) given by the formula ~ ( ~ a , t " ) = ~ ( a ~ ) t " . This action extends to the universal Picard-Vessiot ring R by posing ~(l) = l and ~r(e(g)) = e(r This action commutes with 4). For a difference module M over k the solution space co(M) = ker(~P- 1, R | M) is a C-vector space and is invariant under the action of Gal(C/ko). Using t h a t H ~(Gal(C/ko), Gl(n, C)) is trivial (See [58]) one finds that the k0-vector space co(M) a~(c/ko) has the property C Oko co(M) G'~l(c/k~ --+ co(M) is an isomorphism. In other words r has a natural structure as C| OF(M) where oF(M) is a vector space over k0. We will call this a k0-structure on co(M). Now M ~ oF(M) defines a functor from the category of difference modules over k to the category of vector spaces (of finite dimension) over k0. As in Corollary 6.3 this is an exact, faithful m o r p h i s m of k0-1inear tensor categories. In particular, the Tannakian approach yields a difference Galois group for M , defined over k0, such that G| is the difference Galois group of C(z) | M. T h e ko-structure on oa(M) has also consequences for rationality properties and algorithms concerning difference equations. (Compare [28]).

6.4

A u t o m o r p h i s m s of tile universal P i c a r d - V e s s i o t ring of 7)

For a h o m o m o r p h i s m h : G --+ C* and a constant c E C one defines the a u t o m o r phism ~r of R over 7), commuting with 4), by c~(e(g)) = h(g)e(g) and ~r(l) = 1+ c. This describes the group of all such automorphisms. The group is c o m m u t a tive. As group scheme over C it has the coordinate ring C[e(g)]9 C g][1] with co-multiplication m given by ra(e(g)) = e(g) r,~ e(g) and re(l) = l | 1 + 1 0 1.

CHAPTER 6. DIFFERENCE MODULES OVER P

66

6.5

Difference equations formal Galois group.

over C((z-1))

and the

The Galois group of a difference equation over the field C((z-1)) will be called the formal Galois group. In order to find this group we want to find first the universal Picard-Vessiot ring f / o f C((z -~ )). The ring f/ lies in R, the universal Picard-Vessiot ring of 7). It is clear that f~ must contain 'P a n d / . Any 9 E ~ lies in some finite extension of C((z-X)) and has finitely many conjugates. One can then construct a difference equation over C((z-1)) with eigenvalues the coNugates of 9. This shows that f / m u s t coincide with R. The group of automorphisms of/~ over C((z-1)), commuting with 0, is denoted by Aut(R/C((z -1)), r One finds a split exact sequence of groups

1

1

Any automorphism c~ ofT) over C((z -~ )) acts on g. The natural extension of~r to an element of Aut(R/C((z-l)), r is given by (r(l) = 1 and (r(e(g)) = e((rg). This formula defines the splitting. The group Aut (R/C((z- 1)), r is not commutative. We will describe the Galois group of a difference module (or equation) M over C((z-1)) in case C = C is the field of complex numbers. The solution space of this difference module is V := ker(O~- 1, Rr M). The automorphism group Aut(R/C((z-X)),r acts on V and the image of this group is the formal Galois group of M. One can make this more explicit, by defining a special element called the formal m.onodromy. This 7 acts on 7 ) / C ( ( z - t ) ) by 7(z ~) = e2~iXzx for all A E Q. The action of 7 on ~ is induced by its action on 7 ~*. Finally, 7 acts on R by 7e(g) = e(Tg) and 7(l) = l + 2rci. The image of 7 in Gl(V) will be denoted by 7v and will be called the formal monodromy of the differential module M. The group Aut(t~/7)[1], r and its irnage T in Gl(V) will be called the exponen-

? E AuI(R/C((z-1)), r

tial torus. The group generated by 7 and Aut(R/7~[l], r is Zariski-dense in Aut(R/C((z-*)), r This has as consequence that the Zariski closure of the subgroup of Gl(V), generated by 7v and 7- is equal to the formal Galois group of M. This is completely analogous to the differential case: "The formal differential Galois group of a differential module is generated as algebraic group by the exponential torus and the formal monodronay." (See [40] ). One can use the method of [51] to translate the classification of difference modules over C((z-~)) and its formal Galois group in terms of linear algebra. This works as follows: One considers a category 5r with objects of the form (I/, @ ~ , 7v), where: (a) V is a finite dimensional vector space over C.

6.s

THE FORMAL GALOIS GI~OUP

67

(b) V has a direct sum decomposition V = %~c_c Vy. (e) 7v E GL(V) is supposed to satisfy: 1. 7v(Vg) = VTg, where 7 e Aut(I~/C((z-1)), 49) is the formal monodromy. 2. Let d > 1 be the smallest integer such that 7dq = .q holds for all 9 r with Vg r 0. Then 7r is unipotent. A m o r p h i s m A : (V,q~Vg,Tv) -+ (W,(aoWg,Tw) is a C-linear m a p A : V --+ W such that A(Vg) C Wg for all.q ~ G and 7wA = ATv. The category b r h a s a natural structure as Tannakian category. The forgetful functor cv : (V, @Vg, 7 v ) ~-+ V from 5c to Vectc is a fibre functor. Thus .T is a neutral Tannakian category. For a fixed object (V, qg!@Tv ) of 3r, one considers the full Tannakian subcategory generated by this object. The Tannakian group of the object (V, V~,<5...r Vg~, 7 v ) , i.e., the group scheme corresponding to this subcategory, is easily seen to be the algebraic subgroup of Gl(V) generated (as algebraic group) by 7v and by the group of the linear maps @~=l.Xiidvg, with A1,...,A~ E C* satisfying: If91 " ". . ....g, .. '~ . . . ~ = 1 E C * . = 1 i n ~ then A~ To a difference module M over C((z)) we associate ('~< %Vg, 7v) defined by ~,'~ = Y N ( S o [ / ] e ( g ) ' ~ M ) a n d T v is the action of 7 to V. It can be seen that this induces an equivalence fl'om the category of the difference modules over C ( ( z - I ) ) to the category Y. This equivalence preserves tensor products and is an eqniva.lem-e of Tannakim~ categories. In particular, the formal Galois group of a difference module M coincides with tile Tannakian group of the associated (V, "~ w V'~,% ~). In other words, the formal Galois group is, as algebraic group, generated by the formal m o n o d r o m y 7v and the group @~=lAiidv~, with A1, ...,)~ C C" satisfying: ifg~...9's ~" = 1 in g then

V=w(M)=ker(d~-I,R@M);

= 1 e c-.

The last group is the exponential torus, as defined above. Strictly speaking T is not always an algebraic t.orus since T need not be connected. The Galois group (7 of M is easily seen to have the following properties:

1. GIG ~ is commutative and generated by at. most two elements. 2. There exists a B E Gl(V) such that G ~ is t.he Zariski-closure of the group generated by /3. We note t h a t the term formal monodromy above is appropriate. Indeed, M has a fundamental matrix Y with coefficients in R. Tile elements e(g) and l of R have an interpretation as multivalued functions. Hence Y can be interpreted as a matrix whose coefficients are combinations of functions in C ( ( z - 1 ) ) and F "x, c z , z ~~ e x p ( ~ 0 < ~ < l a~z"), Io9 z. Analyt.ic continuation on a circle around oc coincides with the formal monodromy.

Chapter 7

C l a s s i f i c a t i o n and c a n o n i c a l forms 7.1

A classification

of singularities

The singularity at infinity of a difference equation y(z + 1) = A y(z) over k, where k is one of the fields C(z), kco := C({z-1}) or koo := C ( ( z - 1 ) ) , will be classified in a rough way by the terms: regular, regular singular, very mild, mild and wild. The terminology reflects the asymptotic theory of the equation. The classification can be read of from the formal (i.e. over C ( ( z - 1 ) ) ) classification of the equation. We recall that this formal classification involves the subgroup Q of 72*, where 72 is the algebraic closure of ]r consisting of the elements

z~c exp(r

- q)(1 + z-l) ao

with A E Q, c E C*, a0 E L C C and q is a finite sum ~ 0 < u < l a, z€ with az C C and # C Q. The term L denotes a Q-linear vector space of C such that L ~ ) Q = C. The elements of ~ occuring in the formal decomposition of the equation (over the algebraic closure 7) of ]%o) are called the eigenvalues of the equation (see Theorem 6.2). The equation is called wild if an eigenvalue with A # 0 is present and mild if this is not the case. The equation is very mild if for every eigenvalue one has A -- 0 and e = 1. If the only eigenvalues are oft.he form (1 + z - l ) a~ then the equation is called regular singular. Finally the equation is called regular if the equation is trivial over ]r L e m m a 7.1 The following properties of y(z+ 1) = A y(z) over k are equivalent.

1. 7'he equation is regular.

72

C H A P T E R 7. C L A S S I F I C A T I O N A N D C A N O N I C A L F O R M S 2. There is a formal fundamental matrix F = F(z) E Gl(n, koo), i.e F ( z + 1) = AF(z). 3. There is an equivalent equation v(z + 1) = A v(z) with A := B ( z + 1 ) - l A B ( z ) and B E Gl(n, k) such that the expansion of ft at infinity reads 1 + A2z -~ + Aaz -3 + ....

Proof." (1)~=~(2) is obvious. Suppose that there is a formal fundamental matrix F. One can approximate F by some B E Gl(n,k) such that A := B ( z + 1 ) - l A B ( z ) has the required form. This shows (2)~(3). If A has the expansion 1 + A2z -2 + A3z -a + ... at infinity then one easily shows that there is a unique formal matrix F of the form 1 + F1 z - x + F, z - : + . . . such that F(z + 1) = A F ( z ) . This proves (3)=>(2). II The difference ring kc~[{Za}aEc,l] is defined by the relations Z ~ Z b = Z ~+b, Z 1 = z E k ~ and the action o f r with r ~) = ( l + z - 1 ) ~ Z a ; r =/+log(l+ z-Z). This ring is in fact a subring of the universal Picard-Vessiot ring defined in Section 5.2. L e m m a 7.2 The following properties of y( z + 1) = A 9( z ) over k are equivalent. 1. The equation is regular singular. 2. The equation is trivial over the ring koz[{Za}aec, l]. 3. There is a fundamental matrix with coefficients in koo[{Za}aec, l]. 4. There is a formal F E Gl(n, koc) and a constant matrix D such that F ( z + 1 ) - I A F ( z ) = (1 + z - l ) D. 5. There is an equivalent equation v(z + 1) = A v(z) with := B ( z + 1 ) - l A B ( z ) and B E Gl(n,k) such that the expansion of A at infinity is 1 + A l z -1 + ~4~z-~ + .... P r o o f i Most of the implications follow from the formal classification of difference modules. We will show here (as an example) that (5) implies (4). We may suppose that A has already the expansion 1 + A l z -1 + ... at oc. The matrix A1 is supposed to be in Jordan normal form. Let a Jordan block with eigenvalue A be given. The matrix B is supposed to be a diagonal matrix with z's on the diagonal of this Jordan block and l's on the diagonal of the other blocks. Let A = B ( z + 1 ) - l A B ( z ) = 1 + Alz -1 + .... Then A~ is almost the same as A1. The only change is that A is replaced by A - 1. With operations of this type one can transform the equation to a new equation, again called y(z + 1) = A y(z), such that every eigenvalue A of A1 satisfies 0 < Re(A) < 1. In particular, the differences of distinct eigenvalues of A1 are not integral.

7.1. A CLASSIFICATION OF SING ULARITIES

73

This last property is used in the proof that there exists a unique formal

F = 1 + F~z -~ + F2z -2 + ... such that F(z + 1)-~AF(z) = (1 + z - l ) D with D=A1. 1 Recall that the algebraic closure of ]%0 is denoted by P. This is the field of formal Puiseux series over C in the variable z -1. The following two lemmas are also easy consequences from the formal classification of difference equations. L e m m a 7.3 The following conditions are equivalent

1. The equation y(z + 1) = A y(z) is very mild. 2. The equation is equivalent (over k) with an equation v ( z + 1) = 4 v(z) with expansion A = 4o + 41z -1 + ... at infinity such that 4o - 1 is nilpotent. L e m m a 7.4 The following conditions are equivalent

1. The equation y(z + 1) = A y(z) is mild. 9. The equation is equivalent (over k) with an equation v(z + 1) = 4 v(z) with expansion 4 = A0 + A l z - 1 + . . . at infinity such that Ao is invertible. 3. There is a fundamental matrix with coefficients in 7)[{e(g) }geOm,,~, 1], where O,~ad denotes the subgroup of ~ consisting of the elements {cexp(r

+ z-1)a~

ao 6 L, q=

E

auz"}.

0<#
Over the field k~ one can compare differential modules and difference modules. A differential module M is a vector space over 1r of finite dimension, together with a C-linear action of ~ on M, satisfying the formula d ( f r n ) =

f ~ ( r n ) + ( ~ ) m for f 6 ko~ and m C M. The well known formal classification of differential modules over the algebraic closure 7) of ] ~ can be formulated as follows: One considers Q, the group (or vector space over Q), defined as

{Eat, zF'I finite

sums ,# 6 Q, a, E C}/Q.

t~>o Let L denote a Q-vector space with L 9 Q = C. Then we may identify Q with the space

{~-~a.z'l ~_>o

finite sums ,p 6 Q, a , 6 C, ao E L}.

CHAPTER 7. CLASSIFICATION AND CANONICAL FORMS

74

For q E Q, one defines the one dimensional differential m o d u l e / ) ( q ) = 79eq over 79 by d e q = z-lqeq. For a differential module M over 79 there is a unique decomposition M = OqEQ:D(q)| M(q) where the M(q) are nilpotent differential modules over 79. The set {q E QI M(q) r 0} is called the set of eigenvalues of M . A differential module M over ]~c~ is called mild if the eigenvalues of 79 | M are contained in

{ E

a"z'l finite sums ,# E Q, a~ E C, a0 E L}.

0<#<1

M is called very mild is this set of eigenvalues is contained in

{ E

a"z"l finite sums ,p E Q, a , E C, a0 E L}.

0<_,<1 A lattice M0 C M is a free module over the ring of integers C[[z-~]] of ~:oo such t h a t the natural m a p ]%o | J%o --+ M is an i s o m o r p h i s m One can show t h a t M is mild if and only if there is a lattice M0 with d M 0 C M0. T h e translation into matrix differential equations is: M is mild if and only M corresponds to an equation ~ y = A y where A has expansion Ao + A l z -1 + ... at infinity. Furthermore "M very mild" is equivalent with the existence of a lattice M0, such that d M 0 C M0 and d acts as a nilpotent m a p on Mo/z-lMo. The translation into matrix differential equations is: M is very mild if and only M corresponds to an equation ~ y = A y where A has expansion Ao + A l z -1 + . . . at infinity with A0 nilpotent. On a mild differential module M one defines the operator ~ as the infinite sum exp(~) ~ V" • V~9 It can be seen, by using the topology of C, t h a t this Z_,n!~,dz] infinite expression makes sense and defines a difference module. This difference module is mild. Using the formal classification of difference modules one easily sees t h a t every mild difference module is obtained in this way from a mild differential module. However this mild differential module is far from unique. The situation is more transparent for very mild difference modules. For such a module one easily shows that - d- " - . - l o g ( ~ )

dz

:= E n>l

-( --1 )( "(+ I1 > -

n

1) n

converges on M and defines on M the structure of a very mild differential module. T h e construction above gives an equivalence between the abelian tensor categories of very mild difference modules and very mild differential modules. The equivalence maps regular (regular singular resp.) difference modules to regular (regular singular resp.) differential modules.

7.2. CANONICAL FORMS

7.2

75

Canonical forms

A difference e q u a t i o n y(z + 1) = A y(z) can f o r m a l l y be t r a n s f o r m e d into a canonical form A ~ = F(z + 1 ) - I A F ( z ) . Usually F is t a k e n to be an invertible m a t r i x with coefficients in P and the canonical form A ~ is a direct s u m of blocks g(1 + M g l o g ( 1 + z - l ) ) where g runs over a finite subset of G and with each My a c o n s t a n t n i l p o t e n t m a t r i x . For our purposes this is convenient a n d sufficient. However the finite extension of k ~ needed for F a n d A ~ m i g h t cause p r o b l e m s . In this section we will derive a canonical form and a t r a n s f o r m a t i o n which are defined over koo and koo. We fix an element g0 C ~. In general the one d i m e n s i o n a l m o d u l e with g e n e r a t o r e(go) a n d r e l a t i o n r = goe(go) is not defined over k ~ since go lies in a finite extension of koo. Let T be the smallest field extension of ko~ such t h a t 90 r T. T h e n Te(9o) is a well defined difference m o d u l e over T. Let gi, i = 0, 1, . . . , m - 1 d e n o t e the c o n j u g a t e s of g0 in T. By m i n i m a l i t y m =- [T : k ~ ] . C o n s i d e r now the difference m o d u l e M := Te(go) + ... + Te(g,~_~). T h e g e n e r a t o r of the Galois g r o u p of T / k ~ will be d e n o t e d by 7. We define 7, as usual, by 7 ( z 1/'~) = e'~i/mz 1/m. We m a y s u p p o s e t h a t ")'J(g0) = gj. We let 3' act on M by the p r e s c r i b e d a c t i o n on T and7(e(gi)) = e(gi+l). For convenience we have i n t r o d u c e d the n o t a t i o n gi+krn = gi for 0 <_ i < m - 1 and any k. T h e action of 7 c o m m u t e s with the o p e r a t i o n of (I) on M . Let E(9~0) d e n o t e the set of 7 - i n v a r i a n t e l e m e n t s of M . T h i s is a vector space over ko~ and also a difference module. A n e l e m e n t rrt--1 ~ j = 0 aje(gj), with aj E T, is invariant under 7 if and only if a j + l = ~/aj for all j . P u t r = e 2~i/'~. A n explicit basis of E ( j 0 ) over k ~ is

z klm ~

(kJe(gj) for k = 0 . . . . . m -

1.

j=0

T h i s shows t h a t T | E(9~0) -~ M is an i s o m o r p h i s m . T h e m a t r i x of 9 on this basis has coefficients in koo. We use the n o t a t i o n ~ for g E ~ to d e n o t e an element of the o r b i t of g u n d e r the action of the Galois g r o u p of the a l g e b r a i c closure of ko~ over k ~ . A canonical difference module over k~ if defined to be a direct s u m (over the o r b i t s in G), | | M~ where each M~ is a u n i p o t e n t difference m o d u l e (i.e. with a m a t r i x of the sort 1 + z - i N for some n i l p o t e n t m a t r i x N ) . It is clear t h a t every difference m o d u l e over koo is formally, i.e. over ko~, equivalent with a unique canonical difference m o d u l e over koo. One can t r a n s l a t e this in t e r m s of matrices. Every canonical difference m o d u l e over koo c o r r e s p o n d s to an invertible m a t r i x A c with coefficients in koo. T h i s m a t r i x can be c o m p u t e d by using the given basis of E(t~) a n d by choosing a s t a n d a r d form for n i l p o t e n t matrices. For every difference e q u a t i o n y(z + 1) = A y(z) there is a unique canonical m a t r i x A c a n d a unique invertible m a t r i x F with coefficients in / ~ such t h a t F(z + 1 ) - I A F ( z ) = A ~.

76

C HAP TE R 7. CLASSIFICATION AND CANONICAL FORMS

One might hope that for difference equations over C(z) there is also a "canonical form" defined over C(z) which is over k ~ equivalent with the original one. However the canonical difference modules E(~) considered above are, in general, not equivalent over koo with modules defined over C(z). (See examples 9.8 and 9.9). A possibility would be to replace every E(~) by some module E'(~) which is now defined over C(z) and is over ~'oo equivalent with E(~). A truncation of E(.~) has this property. There seems to be no evident choice for the E'(~). Unlike the case of differential modules, truncations of s with respect to high orders of z -1 are not isomorphic over koo (Over ]%o they are isomorphic). In particular, the asymptotic theories, corresponding to different choices of truncations, are different.

Chapter 8

Semi-regular difference equations 8.1

Introduction

The asymptotic theory of general difference equations has some analogies with the theory for differential equations. However, difference equations are more difficult to handle. In later sections we will study the asymptotic theory, connection matrices and the relations with the difference Galois groups for general difference equations. As a guide to this we will work out this theory for special equations, the semi-regular difference equations. For those equations one can construct asymptotic lifts "by hand". Moreover those lifts are unique and are defined on large sectors. The relation with difference Galois groups is rather transparent for semi-regular equations. A difference equation over koo (or over C(z) or koo ) is called semi-regular of type n if the equation is formally equivalent (i.e. over/~:o~ ) to a matrix equation y(z + 1) = A y(z) with a diagonal matrix A, having nth roots of unity on the diagonal. The following properties are equivalent: (1) The equation is semi-regular of type n. (2) The "eigenvalues" of the difference equation are contained in #,~ := the group of the nth roots of unity in C*. (3) Tile equation has a full set of solutions in the difference ring k~[e(c)]ce... We will write this ring as koo[/t~].

78

CHAPTER

8. S E M I - R E G U L A R

DIFFERENC, E E Q U A T I O N S

The notion of "semi-regular of type 1" coincides with "regular". A difference module is called semi-regular if it. is semi-regular of type n for some n > 1. The semi-regular (of type n) difference modules over ko, form a full subcategory of the category of all difference modules over ko~. The tensor product of two semiregular modules (of type n) is again semi-regular (of type n). Further quotient modules and submodules of a semi-regular module (of type n) are semi-regular (of type n).

8.2

S o m e easy a s y m p t o t i c s

We recall the definition of asymptotic expansion. Let. a sector at ~ be defined by {z E C[ [z[ > R, arg(z) E (a,b)}. A holomorphic function f on this sector has the asymptotic expansion ~,~>a a~ z - ~ E koo = C ( ( z - i ) ) if there exists for every N > A and every positive e a constant C.' > 0 such that

If(z)- ~

a~z-~l < Clzl - x

A<_n
holds for all z with ]z] _> R + e and arg(z) E (a + e,b - e). The first lemma that we consider is rather elementary. This l e m m a suffices for the construction of the connection matrix of a semi-regular equation. We consider ordinary difference equations in the neighborhood of oo. As before, the field of convergent Laurent series at infinite is denoted by ko. = C ( { z - 1 } ) and the field of formal Laurent series in z - i is denoted by ko~. A subset of the complex plane of the form

{~ c cl lib.(z); > R or (l~l > R and R,~(z) > 0)} will be called a right domain. A subset, U of C is called a left domain if - U is a right domain, L e m m a 8.1 Let 9 be a formal solution (i.e. with coefficients in koo ) of the equatwn y(z + ]) = A y(z) + a, where A and a have coefficients in koo. The expansion of A at oo is supposed to have the form A0(1 + A i z -1 + A2z -2 + ...) whe~v Ao is a semi-simple matrix .such that all its eggenvalues have absolute value I. There exists a unique meromorphic vector yright, such that: (1) ?]right is defined on a right-domain. (2) Y~ight has 9 as asymptotic ez'pansion at ~ in a sector with arg(z) E (-~r + ~, +r~ - c) for every ~ > O. ('~) Yridht(Z + l) -: A Yright(Z) §

8.2. S O M E E A S Y A S Y M P T O T I C S

79

P r o o f : T h e n o r m s t h a t we will use for m a t r i c e s M and vectors v are the usual ones, n a m e l y ILM[I = x / E I~/r 2 and II~ll = v~-Iv~l 2, We have chosen coordinates such t h a t A0 is a diagonal m a t r i x . T h e n o r m of A0 is therefore 1. Choose some m e r o m o r p h i c vector 9x which has a s y m p t o t i c e x p a n s i o n 9 on any sector at oc with o p e n i n g ( - r r + e, ~ r - e) for every e > 0. P u t Y2 = 9I + 9 , where 9 should be chosen such t h a t 9 has a s y m p t o t i c e x p a n s i o n 0 at the s a m e sectors at oc a n d such that, y2(z + 1) = A y2(z) + a. T h i s m e a n s t h a t 9 should be a s o l u t i o n of

g(z + ~) - a ( ~ ) g ( ~ ) = - ( y ~ ( ~ + ~) - A ( - - ) > (~) - ~) := - b ( ~ ) , where (by c o n s t r u c t i o n ) b(z) is a m e r o m o r p h i c vector which has a s y m p t o t i c e x p a n s i o n 0 in the s a m e sectors at oc. One proposes the solution OG

#(z) = ~

A(:)-IA(~

+ 1) -~ . . . . 4(~ + 7 ~ ) - ~ b ( : + 7~).

~l=0

C o n s i d e r a b o u n d e d open sector S C C with arg(z) E ( - r e + e, rr- e). On S one has the following inequalities: (1) T h e r e is a c o n s t a n t c > 0 such t h a t for all z E ,-g one has c m a x ( l z l , n ) _< Iz + zzI < 2 m a x ( l z I,n)

(2) IIAoA(:)-lll _< (1 + dlzl-*) for some c o n s t a n t d > O.

(3) Ila(~-)ll < ~

for all ~ =

1,2,...

and some constants

Cs.

Take s > > 0. We then have

D)...(I+

IIA(~) -* ...A(~-+~)-lb(~+,~)ll <_ (1+ I~1

D )

I~l+'~

C,

Iz+np'

where D = [2dc -1] + 1. T h e e s t i m a t e in this f o r m u l a is equal to n+l,

(1 + ~ 7 - / . . . ( 1

+

n + 1

Izl+D-1

C'~ ) Iz+,~p

We m a y assume t h a t ]z I >_ 1 holds on the sector ,~. Using this and the i n e q u a l i t y ([) one finds an est.imate

iiA(z)-~ ...g(z + n)-~b(z + n)ll ~ (:~e-,(" + l) D Tile infinite s u m therefore converges uniformly on S. estimate

7z=l

F u r t h e r one finds the

80

C H A P T E R 8. S E M I - R E G U L A R D I F F E R E N C E E Q U A T I O N S

This shows that g has asymptotic expansion 0 in S and therefore also on the sectors with arg(z) E (-re + e, re - e). Finally, by construction g satisfies the required equation. T h e vector Y2 exists on some sector at ~ with opening (-Tr, re) and satisfies conclusions (2) and (3). The equation 3. shows that y: has an analytic continuation Yright on a right-domain. Suppose t h a t Y~.ight also satisfies conclusions (1), (2) and (3). The difference V ofyright and y'right has asymptotic expansion 0. The formula v ( z + 1) = A v(z) implies v(z) = A ( z ) - l . . . A ( z + n ) - l v ( z + n). For a fixed z, any n _> 1 and any E s > D + 2 one finds estimates Iv(z)l <_ I~+,I,-D_~ for some constant E > 0. This shows t h a t v = 0. We note t h a t the point of departure in Birkhoff's paper [9] is a difference equation of the form y(z + 1) = zUA0(1 + A l z - I + . . . ) y ( z ) with A0 an invertible m a t r i x such that all the eigenvalues are distinct. Birkhoff claims to find a right lift Fright and a left. lift FleJt of the symbolic fundamental m a t r i x with -1 F left good asymptotic properties. The connection matrix S is defined by ]'right and Birkhoff proves a number of statements about S. If the eigenvalues of Ao have absolute value 1, then L e m m a 8.1 confirms this because the term z ' is not essential. If the eigenvalues of A0 do not. all have absolute value 1 then some statements of [9] are no longer valid.

8.3

T h e c o n n e c t i o n matrix of a senti-regular equation

We begin with some comments and notations. (1) The ring of meromorphic functions, defined on a right domain and having an asymptotic expansion in ko~ on the sectors at oc, with opening ( - r r + e, re + e) for some positive e, is denoted by Ri. This is a difference ring containing the field of convergent Laurent series ko~. One defines in a similar way a difference ring L e of meromorphic functions on left domains in C. In fact, f ( z ) belongs to Le if and only if f ( - z ) belongs to Ri. In other words, the functions in L e are defined on a left domain and have asymptotic expansion in koo on the sectors which are given, with some abuse of notation, by (e, 2re - e) for every e > 0. A variation on lemma 8.1 is:

There exists a unique vector Yteft with coordinates in Le, such that the asymptotic expanszon of yleft is 9 and such that ylejt(z + 1) = A ylr + a holds.

8.3.

THE CONNECTION

MATRIX

81

(2) An upper half plane is a subset of C of the form {z 6 C[ Ira(z) > c} for some positive real number c. By U p we denote the set of meromorphic functions, defined on some upper half plane and having an asymptotic expansion in kco. The last statement means that this asymptotic expansion holds for every sector with arg(z) E (~, rr - ~) with c > 0. The space of meromorphic functions, defined on some lower half plane and with asymptotic expansion in k~ is denoted by Lo. Its definition is similar to that of Up. (3) Let M be a semi-regular difference module of type n over k~. We consider solutions of this difference equation with coordinates in various difference rings. The first ring is k~[#,d. Put, ;o(M)

= k ~ , , ( ~ - 1,

k~[,~] o M).

The set of 4)-invariant elements of koo [/J~] is equal to C, because it is a subring of the universal ring introduced in Section 6.2. Hence a 0 ( M ) is a vector space over C and its dimension over C is equal to the dimension of M over kcr Moreover, the canonical map

koo~]

|

~o(M) -+

koo[/1,~]<~kooM

is an isomorphism. (4) M denotes again a semi-regular difference module of type n over k~. We want to define wright(M), which should be the space of solutions of M living above a right domain. For this we introduce the difference ring Ri[#, d := Ri[e(c)]cr

.

We do not give the symbols e(c), c E I1~ an interpretation as functions on a right domain! At this point we deviate from the current practice for difference equations. An interpretation of the symbols e(c), c E # ~ as functions on some domain of C
C H A P T E R 8. S E M I - R E G U L A R D I F F E R E N C E E Q U A T I O N S

82

the C-action. Similarly one defines a Cn action on Le[t,~], Up[#n], L o [ # , ] . We introduce now

coright(M ) = k e r ( d P - 1, Ri[/~,~] ~"k~ M), and similarly W*eyt(M) = k e r ( r

1,Le[p,~] & ~ . M)

wupper(M ) = lee,'(~ - 1, Up[/~,z] ?)k~ M) and W~ow~(M) = k e r ( ~ - 1, Lo[#,~] Ci:~k~ M). Each of these spaces has a C~-action. There are obvious, Cn-equivariant, m a p s from w~ight(M) and ccl
The same

(2) The set ofO-inL, ariant element.s of Ut,[#~] is C{~L,,} where u = e '~i~ and u~ = u. The C,~-action on C{u,~} coincides with the action of the Galois group of the field extension C({u.,~}) D C({u}). (3) The set of ~-invariant elem.ents of Lo[#~] is C { u 2 1 } with the same action

Of C~. Proof." We consider a r element f of Up[,u,~]. There is a unique expression f = ~ e , , fce(c) with all J'~ E U p . Then ~(f~)c = f~ and so f~ = e2~iX~h(u) with A E 1 / n Z , 0 < A < l, c = e -2~ia, u = e 27riz and h a holomorphie function of u for 0 < lug- < e for some positive c. T h e function f~ has an a s y m p t o t i c expansion 9c C k ~ . From o(9c)c = 9~ one concludes t h a t g~ is zero if c 7~ 1 and g~ E C if c = 1. One finds the necessary and sufficient, condition h E C{u}. T h e set of 0-invariant elements is therefore equal to the ring C{a}[ta], where ta = e~e~iX~e(e- ~ i x ) and where A runs over the set. [0, 1)N 1 / n Z . T h e structure of this ring is found by making a small c o m p u t a t i o n . T h e element tl/,~ is seen to satisfy the equation t'~/,~ = u. This equation is irreducible over the field C({u}) and the subring C{t,/,~}, of tile ring of all invariants, is isomorphic to the integral closure of C {u} in the unique field extension of degree n of C({u}). With the notation u,~ = tl/,,, the proof of part 2. is complete. T h e proof of part 3. is similar. A r element of Ri[l,,~] is a sum of o-invariant d e m e n t s fee(c), with c E /z,, and J'~ E Ri. As before, we can write J~. = e>r~X~h(u). For e = 1, the condition that f~ has an a s y m p t o t i c expansion implies that h is constant. For c 5s 1, tile a s y m p t o t i c expansion of f~ is 0 and this implies that h = 0. This proves the first part. of 1. T h e proof of tile second part of 1. is similar. |

8.3.

THE CONNECTION MATRIX

83

It is confusing to identify the element u,~, defined in the proof of L e m m a 8.2, with u 1/n = e 2~i~/~. The action of 0 is trivial on u~, but the action of 0 is not trivial on u~/'~! In the work of Birkhoff and others on difference equations this confusing notation is present.

Proposition 8.3

Let M

be a semi-regular diffe~vnce equation of type n over

k~. (1) There are canonical isomorphisms wo(M) + w'rOht(M) and vao(M) -+ ~ol~ft(M). (2) There are canonical isomorphisms

c { ~ } o~

~

-+

~ p , ~ ( M ) and C { ~ } G~ <~S~ -~ ~ ( M ) .

(3) There are canonical isomorphisms

C{ttn 1} @C CO'right -'+ ~lower(M)

and

C{ltn 1} @c OOleft --+ r

Proof." 1. Write f E w0(M) as a su,n f = ~ f ~ e ( c ) with c E #,~ and fc E k ~ 0 M. The "coordinates" {f~} of f satisfy a semi-regular equation over k ~ . According to the Lemma 8.1 there are unique lifts fright, c E Ri. The element f~i~ht := Y-~ fright, ce(c) lies in a;~i~l~t(M). This defines the first m a p of part (1) of the proposition. Since the set of constants of Ri[#,~] is C, the m a p is an isomorphism. We note that there is an obvious surjective homomorphism of difference rings Ri[#,~] -+ k~[#,~]. Using this map, one also finds a C-linear m a p ~right(M) ~ ~ o ( M ) . This is the inverse of the first map. The other part of 1. has a similar proof. 2. The m a p w,.ight(M) --+ cZ~ppe~(M ) is induced by the obvious map of difference rings Ri[#~] --+ Up[#,~]. After taking the tensor product with C{u,~} one obtains an isomorphism since C{u~} is the set. of r elements of Up[tt~]. The proof of the Proposition is now clear. The term "canonical" means that the maps are functorial in M and conmmte with tensor products. This follows from the unzqueness of the asymptotic lift in Lemma 8.1. | There are now two isomorphisms

We define an upper connection map SM, upper C Aut(C{un} ~C wo(M)) by first taking the "left" map and then the inverse of the "right" map. The definition of the lower connection map SM, lower iS similar. This map has coefficients in C{u~*}. Tile maps SM, ~p;~,. and SM, lo..... are C,~-equivariant. Moreover

"

84

C H A P T E R 8. S E M I - R E G U L A R DIFFERENCE E Q U A T I O N S

5"M, upvr(O) and 5'M, l ...... (oo) are the identity on w0(M). The aim is to formulate in a precise way that a semi-regular difference module M over koo, of type n, is determined by the vector space ,a0(M) with its C~action and the two Cn-equivariant isomorphisms 5'M, u p p e r , 5"M, lower. In particular, the difference Galois group of M should be expressed in those items. Another important point is whether one can prescribe the connection maps 5"M, upper and SM. ~o*~er. The tool for the last. question is a theorem on asymptotic expansions of matrices, due to Malgrange and Sibuya.

8.4

The t h e o r e m of Malgrange and Sibuya

In this section we discuss this theorem and adapt it. for later use. Let 5'1 denote the circle of directions at oo. On this circle one considers a sheaf A. This sheaf is defined by: A(U) is the ring of (germs of) merolnorphic functions living on a sector corresponding with the interval U of 5'1 having an asymptotic expansion in koo on the sector. Let A ~ denote the subsheaf of A consisting of the meromorphic functions which have asymptotic expansion 0. On the space 5'5 there is an exact sequence of sheaves

O-+ A~ --+ A --+ ir

--+ O.

The last sheaf is the constant sheaf on 5'1 with stalk koo. The exactness is a translation of the well known fact that every formal Laurent series is the asymptotic expansion of a nleromorphic function on any proper sector at oz. The cohomology sequence is 0 -+ 0 -+ koo ~ kco --+ H I ( S I , A ~ --~ -.. A special case of the theorem of Malgrange and Sibuya states that the map ~'~ --+ HI(5'I,.A ~ is surjective. To illustrate this, consider the two sectors $1 and $2 given by arg(z) :fi rr and arg(z) 7~ 0. The intersection S1NS2 is the union of two intervals or sectors. An element F C L'~o can be lifted to elements F1, F_~ on the two sectors $1 and S~. Their difference - F 1 + F2 lies in A ~ and can be seen as a 1-cocycle for the group HI(S1,A~ This describes the map ko, --+ H I ( S 1 , A ~ The surjectivity of this map means the following. Let. an element f E A~ A $2) be given. This is a 1-cocycle representing an element of H I ( S 1 , A ~ Let F E k~ have image ~. Then the lifts F1, F.e on S1,$9_ have the property that - F x + F2 is equivalent to the 1-cocycle f. One is allowed to change F1 and F9 by elements in A~ and A~ After this change one finds an equality - F 1 + F.~ = f. Another formal Laurent series G with image differs from F by a convergent Laurent series. The Malgrange-Sibuya theorem (see [61] for an exposition of that theorem) says that a similar result holds for

8.5.

R E G U L A R DIFFERENCE EQUATIONS

85

Gl(n, ~:oo) in the place of ]%0. One has again an exact sequence of non abelian groups 1 --~ H --~ G l ( n , A ) -+ Gl(n,]%o) --+ 1,

where H denotes the subsheaf of Gl(n, .A) consisting of the matrices with asymptotic expansion 1. The corresponding cohomology sequence reads

1 -+ 1 -+ G l ( n , k ~ ) -+ G l ( n , ] ~ ) ~ H I ( S i , H ) ~ ... The H i term is not a group but only a set since it is the cohomology of a sheaf of non abelian groups. The statement of Malgrange and Sibuya is that Gl(n,]%o) ~ H i ( S ~ , H ) is surjective. With the notations Si,S2 above this is equivalent with the following assertion:

Let f E H(S1 N 52), then there exists an element F E Gl(n, ] ~ ) and asymptotic lifts Fi, F~ on the sectors Si, $2 such that F~-IF2 = f . Any other G E Gl(n, ]%,) which maps to the same 1-cocycle has the form G = B F with B E Gl(n, k ~ ) . The following statement is equivalent to the Malgrange-Sibuya theorern but does not use coordinates. Consider a triple (V, A~,AI) of the following data: (a) V is a finite dimensional complex vector space. (b) A~ is an Up-linear automorphism of U p @ V which has the identity on V as asymptotic expansion. (c) At is a Lo-linear automorphism of Lo | V which has the identity on V as asymptotic expansion. Then there is an automorphism F of ]%o | V with F~ight and First as asymptotic lifts. Frigh t is an automorphism of Ri @ V and Fleft is an automorphism of Le | V, such that the "l-coeycle" F~ilhtfleft is equal to the pair (A~,AL). Further F is unique up to multiplication on the left by some automorphism B of k ~ | V. Define M C l%o| as F - i ( k o o | It is clear that M is a vector space over koo and that the canonical m a p ]%0 | M --+ ] ~ | V is an isomorphism. We will call such an M a k~-structure on koo | V. Another choice B F for F does not change M. On the other hand every k~-structure induces a (non unique) F, automorphism of ]%o | V with F ( M ) = koo @ V, and a unique "l-cocycle" --1 F~ightFlest. In this way the collection of the triples above is in natural bijection with the k~-structures on ]%~ | V.

86

8.5

C H A P T E R 8. S E M I - R E G U L A R D I F F E R E N C E E Q U A T I O N S

Regular difference equations

A regular difference module M over koo is the same thing as a semi-regular module of type 1. We have associated to M the object (w0(M), SM, upper, SM, lowe",'). The first result that we will prove is: L e m m a 8.4 Given an ob)ect (V, T~, T~) with:

(1) V is a complex vector space over finite dimension. (2) Tu is a C{u}-linear automorphism of C{u} .@ V such that T~(O) is the identity on V. (3) T~ is a C{u-1}-linear automorphism of C{u -1} O V such that Tt(oo) is the identity on V. Then there is a regular difference module M over k~o and a C-linear isomorphism f : wo(M) -+ V such that f S M , u p p e r = T u f and f S M , l . . . . . = TLf. P r o o f i The object (V, T~, T~) induces an automorphism F of koo | V and a koostructure M C /%o | V. The space ko~ @ V is given the natural C-action where r is the identity on V. The r of (T~,Tt) implies that r = B F for some automorphism B of koo | V. The koo-structure M is invariant under r In particular M is a difference module over k~. It is easily seen that M has the required properties. II We will now formulate in a more precise way that a regular difference equation over koo is characterized by the two connection matrices. For this purpose we introduce a category C, whose objects are the (V, T~,T~) with: (1) V is a complex vector space over finite dimension. (2) T. is a C{u}-linear automorphism of C{u} @ V such that T~(0) is the identity on V. (3) Tt is a C{u-Z}-linear automorphism of C{u -1} ~) V such that Tt(oo) is the identity on V. A morphism f : (V,T,,,TI) --+ ( V ' , T ' , T [ ) is a C-linear map f : V --+ V' such that IT~ = T ' f and fTt = T[f. In the obvious way one can define tensor products, (internal) Horn et cetera in C. In this way C becomes a C-linear tensor category. Let V e c t c denote the tensor category of finite dimensional complex vector spaces. The forgetful functor C --+ Vectc, given by ( V , T , , T I ) ~+ V, is a fibre functor. Thus C is a neutral C-linear Tannakian category. The following theorem contains more or less all the information about regular difference modules over koo.

8.5.

R E G U L A R DIFFERENCE EQUATIONS'

87

T h e o r e m 8.5 (1) The functor F : M ~-+ (wo(M),5"M, upper,5"M, l . . . . ) f r o m the category of the regular difference modules over koo to C is an equivalence of tensor categories.

The difference Galois group of a regular M over k ~ is the smallest algebraic subgroup G of Aut(wo(M)) such that 5"M, ~pp~ E G(C{u}) and 5'M,~. . . . 9 ~ G(C{~-~}). (3) A linear algebraic group G is the difference Galois group of a regular difference equation over koo if and only if G is connected. Proof." (1) The preceding lemma states that every object of C is isomorphic to some F M . This leaves us with proving that the C-linear map Horn(M1, M2) -+ Hom(FM1, FM2) is a bijection. It suffices to take M1 = 1 := the one-dimensional trivial difference module over koo and M2 = M with arbitrary M. The lefthand side is equal to {m E MI r = m}. Since F 1 = (Ce, 1, 1), the righthand side is equal to

{ m e ~oo O Mlr

= m,

5"M, ~pp~,.m = m ,

SM, l....

m = m}.

It is clear that the two sets coincide. (2) An object (V, T~, T~) o r e generates a full tensor subcategory {{(V, T~, Tl)}} of C. The objects of this subcategory are obtained form (V, Tu, Tl) by taking the "constructions of linear algebra", i.e. tensor products, duals, subobjects and quotient objects. The forgetful functor is again a fibre functor and the smaller category is again a neutral Tannakian category. The corresponding group G will be called the Tannakian group of (V,T~,,T~). This group is actually a subgroup of Aut(V) and can be described as follows: A E Aut(V) belongs to G if and only if for every construction c of linear algebra with result (c(V), e(T~), c(Tt)) and every subobject (V', T ' , T~') the space V ' C c(V) is invariant under c(A). It is not difficult to see that G is actually the smallest algebraic group such that T~ E G(C{u}) and Tl E G(C{u-~}). This and (1) proves (2). (3) From (2) and T~ (0) = 1 and Tt (oo) = 1 it follows that G is connected. On the other hand, let a connected linear group G C Aut(V) be given. A variation on proposition 3.3 shows that there is a T~ (with T~(0) = 1) such that G is the smallest algebraic group with T~ C G(C{u}). Then the object (V,T~, 1) has Tannakian group G. According to (1) this finishes the proof of (3). II Suppose now that M is a regular difference module over C(z). Then the right and the left lifts of elements in wo(M) are actually meromorphic on all of C (and have the required asymptotic behavior on the sectors $1 and 5'2). It follows that 5"M,~ppe~ and SM. l o ~ are meromorphic on all of C. Moreover they coincide. Put 5"M = 5"M, uppe~ = 5"M, tower. The coefficients of 5'M are

C H A P T E R 8. SEMI-REGULAR DIFFERENCE EQUATIONS

88

meromorphic functions in u E C*. Further SM(0) = SM(OO) = 1. Hence S M has coefficients in the field C(u). Let. us introduce another category C*, whose objects are the pairs (V,T) with V a finite dimensional complex vector space and T an automorphism of C(u) | V such that T(0) = T(oc) = 1. As before C* is a neutral Tannakian category. In analogy with the last theorem one can prove the following:

(1) The functor F : M ~-4 (~z0(M), SM) form the category of the regular difference modules over C(z) to C* is an equivalence of tensor categories.

C o r o l l a r y 8.6

(2) The difference Galois group of a regular M over C(z) is the smallest algebraic subgroup G of Aut(aoo(M)) such that SM E G(C(u)). (3) A linear algebraic group G is the difference Galois group of a regular difference equation over C(z) if and only if G is connected. For a matrix difference equation y(z + 1) = A y(z) (with module M) over -1 C(z) one can define the connection matrix S a s Frightfleft, where Fright and Ft~yt are the two lifts of a formal fundamental matrix F of the equation. This matrix S is clearly the matrix of the connection map SM with respect to a given basis (depending on the choice of F) ofw0(M). We note that the last corollary has the following consequence: C o r o l l a r y 8.7 Let y(z + 1) = Aiy(z), i = t,2 denote two regular difference

equations over C(z) with connection matrices Si. The two difference equations are equivalent if and only ~f there exists a g E Gl(n, C) such that $1 = gS29 -1 .

8.6

Inverse problems for semi-regular equations

We want to extend the previous results for regular difference equations to the case of semi-regular modules of type n over ko~. For this purpose we introduce a category C~. An object (V,T~,TI) of this category consists of (1) a finite dimensional vector space V over C; (2) an action of C~, the cyclic group of order n, on V; (3) two C,~-equivariant automorphisms T~ of C { u,~ } | V and Tt of C { u~ 1} | V such that T~(0) and T~(oo) are the identity on V. A morphism f : (V, T,,TI) -+ ( V ' , T ' , T { ) i s a C-linear map f : V --9 V' such that T ' f = fT~ and T{f = fT~. The tensor product of two objects (V,T,,T~) and ( V ' , T ' , T { ) is given as (V | V ' , T , | T',T, | T{). The category is easily seen to be a C-linear tensor category. Let Vectc denote, as before, the tensor

8.6.

I N V E R S E PROBLEMS

89

category of the finite dimensional vector spaces over C. The forgetful functor 71 :C~ --+ Vectc, given by (V,T~,tt) ~ V is a fibre functor. This makes C,~ in to a neutral Tannakian category. The main result on the semi-regular modules of type n over koo is the following.

S.S (1) The functor F : M ~-~ (w0(M), SM, upper, SM, l . . . . ) from the category of the semz-regular modules over koo of type n to the category C7~ is an equivalence of tensor categories.

Theorem

(2) The difference Galois group G of a semi-regular difference module M of type n over k ~ is the smallest algebraic subgroup of Aut(wo( M) ) such that (a) G contains the image of Cn. (b) SM, upper e C ( C { u n } ) . (C) SM, I. . . . ~ ~ ( C { u n l } ) . P r o o f : (1) It is rather clear that F is C-linear and commutes with tensor products. In order to show that F is an equivalence we have to show that every object of C,~ is isomorphic to some F ( M ) . Further we have to show that Horn(M1, M2) -+ H o m ( F M 1 , FM2) is an isomorphism of linear vector spaces over C. We start with the latter and observe that it suffices to take M1 -- 1 (i.e. 1 is the trivial difference module of dimension 1) and Ms -- M. The first vector space is then equal to the set of elements rn E M with r -- m. We note that F 1 = (Ce,Tu,Tl) with T~ and Tl the identity. The second vector space is therefore equal to the set of elements v E w0(M) such that v is invariant under C,; SM, upperV = v and SM, ZowerV = v. For such a v the Cn-invariance implies that v C ker(d2 - 1, ]%o | M). The two lifts of this formal solution of the difference equation coincide on the intersection of a right domain and a left domain. It follows that v E M and r = v. This proves the first part. Let an object (V, Tu,Tz) of Cn be given. On the space C[pn] Q V one defines a e-action by the given action on cLu~ ] and the trivial e-action on V. The C,~-action is induced by the canonical C,~-action on C[p,~] and the given action on V. The two actions commute. Put W := (C[#,~] | V) c". This is a complex vector space with trivial C,~-action and (in general) a non trivial e-action. The canonical isomorphism C[#,~] | W --+ C[#,~] | V is equivariant for Cn and r The automorphism T~ of C {u,~ } | V is extended to a C {u,~ } [#,~]-linear automorphism of C { u,~ } [p,] | V = C {u~ } [#~] | W. This extension is again denoted by T, and is equivariant for r and Cn. A small calculation shows that the set of the C,~-invariant elements of C{u,~}[#~] is equal to C{e2~iz/"}. This has as consequence that the set of C.-invariant elements of C{u,~}[p,] | V equals C{e 2~/~/'~} | W. The automorphism induced by T, is denoted by Su and is r In a similar way one defines a r automorphism St of

C{e -2~izl~ } | W.

90

C H A P T E R 8. S E M I - R E G U L A R D I F F E R E N C E E Q U A T I O N S

We apply the theorem of Malgrange and Sibuya to (W, S~, St) and we find a koo-structure M on koo Q W. Since ~g~ and Sz are ~-equivariant the koo-module M is invariant under ~, defined on k"o~@ I47 by the ordinary ~ on the first factor and the action of r on W introduced above. Thus M is a difference module over koo. It is an exercise to verify that F]~4 is isomorphic to (1/, T~,TI). (2) The proof is analogous to the proof given in theorem 8.5.

|

This theorem has an analogue for semi-regular difference modules of type n over C(z). Let M be such a module, then the two connection maps SM, ~pp~ and SM, low~,- are meromorphic on all of C and coincide. Put SM = SM, upper = S M , lower. This is an automorphism of C(u,~) | V such that SM(0) and SM(OO) are the identity on V. Let C~ denote the category with as objects the pairs (V, T) with V a finite dimensional complex vector space and T a C~-equivariant automorphism of C(u~) | V such that T(0) and T(oc) are the identity on V. T h e o r e m 8.9 (1) The functor F : M ~-~ (Wo(M),SM) from the category of the semi-regular modules over C(z) of type n to the category C* is an equivalence of tensor categories.

(2) The difference Galois group G of a semi-regular difference module M over C(z) is the smallest algebraic subgroup of Aut(r such that (a) G contains the image of C~.

(b) S ~ e a(C(~)). The difference Galois group G of a semi-regular equation of type n over koo has the property that G / G ~ is cyclic of an order dividing n. Indeed, let Z denote the smallest algebraic subgroup of Aut(w0(M)) such that SM, ~vp~, C Z(C{u,~}) and SM, lo,u~, E Z ( C { u ~ I } ) . From the condition SM, ~pp~r(O) = SM, low~,(oo) = 1 it follows that Z is connected. The C,~-equivariance of the pair SM, ,,pp~, SM, ~o~,~r implies that the image of C~ in Aut(~00(M)) lies in the normalizer of Z. The group Z has a finite index in the group Z ' generated by Z and the image of C,~. This index is a divisor of n. Hence Z' is algebraic and coincides with G. And so Z = G ~ For a semi-regular difference module M of type n over C(z) with difference Galois group G, the situation is similar. The group G ~ is the smallest algebraic group such that SM E G~

T h e o r e m 8.10 A linear algebraic group G is the difference Galois group of a semi-regular difference equation over koo if and only if G / G ~ is cyclic. P r o o f : One implication is already discussed above. Let G C Aut(V) be given such that G / G ~ is cyclic. It suffices to construct an object (V, T~, T~) of C~ with Tannakian group G.

8.6.

INVERSE PROBLEMS

91

Choose an element d E G which maps to a generator of G / G ~ The unipotent element d~ in the Jordan decomposition d~d~ of d belongs to G ~ Thus d~ is also mapped to a generator of G / G ~ and we may (and will) suppose that d is semi-simple. The commutative algebraic group Z generated by d is a subgroup of G. The group Z ~ is a torus and Z / Z ~ is cyclic. By construction, the map Z / Z ~ -+ G / G ~ is surjective. There is an element h E Z C G, which has roots of unity as eigenvalues and maps to a generator of Z / Z ~ Thus h maps also to a generator of G / G ~ Let the order of h be n. Then the C,~-action on V is given by the homomorphism p : C,~ -+ Aut(V), which sends a generator ~r of C~ to h E G. We will further take T~ = 1. The choice of the automorphism T := T~ of C{u=} | V must be such that (a) T(0) = 1 and G ~ is the smallest algebraic group with T E G ~ (b) T is Cn-equivariant. Condition (b) is a rather complicated one. The action of the generator a of C~ on C{u,~} | V is given by ~r(I | v) = (cr(/)) | h(v). The equivariance of T reads T(cr| h) = (~| h)T. Put ~ = crT~ -1. This is the natural action of o" on the points of G ~ Then ~ = h - l T h . Let O(G ~ denote the coordinate ring of the group C ~ The action of h on G ~ by conjugation induces an action of h on O(G~ The equivariant element T corresponds with an equivariant homomorphism of C-algebras T* : O(G ~ --* C{u,~} such that the preimage of the maximal ideal of C{u,~} is the maximal ideal of the point 1 E G ~ We want to produce such a T* which is also injective. The injectivety of T* means that T ~ Z(C{u~}) for any proper algebraic subset of G ~ Let 06o,1 denote the local ring of G ~ at the point 1 E G ~ . Let t l , . . . t a denote generators of the maximal ideal m, where d be the dimension of G ~ The tj can be chosen such that they map to eigenvectors in m / m 2 for the action of m 2 (with ~ a primitive nth root of unity h. In other words h(tj) E ~nbj tj -t- __ n-1

and 0 _< bj < n). Put sj = ~ k = o ~((kbJhk(tJ)" Then the 8 1 , . . .

,8 d

are again

generators for m. Further h(sj) = ~ J s j for all j. Let 0 ~ 1 denote the analytic local ring of G ~ at the point 1 E G o 9 The S l , . . . , s d form also a basis for the maximal ideal of this local analytic ring. Therefore 0 G~'~~ = C { s , , 9 " " , sd}. There exists a homomorphism of local analytic rings ~ 9 OC~r~ co.1 -+ C{u,~} such ba

that each element sj is mapped into u~ uC{u} and such that the ~(sj) are algebraically independent over C. This ~p is clearly equivariant. The composition

T* : O(G ~ -+ 0~o,1 r C{u,.} is equivariant. Its kernel is 0, since the field of fractions of O(G ~ has transcendence degree d over C. Hence T* produces the required T. |

92

C H A P T E R 8. S E M I - R E G U L A R D I F F E R E N C E E Q U A T I O N S

We would like to extend the last theorem to the case of semi-regular difference equations over C(z). A result, in this direction is:

T h e o r e m 8.11 A linear algebraic group G is the difference Galois group of a semi-regular difference equation over C(z) if

(1) G / G ~ is cyclic. (2) There exists an element of finite order h E G which maps to a generator of G I G ~ and lies in the connected component N ( G ~ ~ of the normalizer N ( G ~ of G ~ Proof." It suffices to construct an object ( V , T ) of C~ for suitable n with Tannakian group G C A u t ( V ) . For n we take the order of h E G and the C~-action is given by: a generator of C~ is m a p p e d to h. The element h is semi-simple and lies therefore in a torus contained in the normalizer of G ~ We may suppose t h a t this torus has dimension one. The C,~-action on V can then be extended to a G,~-action p : G,~ --+ A u t ( V ) with the properties: p(~,) = h and the image p(G,~) lies in the normalizer of G ~ Let E E A u t ( C ( u ~ ) | be the image under p of the element u~ in the group G m ( C ( u ~ ) ) . Then E lies in the normalizer of the group G~ Take any S E G~ T h e n T := E - 1 S E lies in G~ and T is C~-equivariant. Indeed,

aT = ~E - 1 S a E = fl(~n)-l E-1SEfl(s

= h-lTh.

We have to make a special choice for S, in order to obtain a T with the required properties. The element S is seen as a rational map from P ~ to G ~ T h e first condition on S is t h a t the matrix S - 1 has zeroes of sufficiently high order at u = 0 and u = oe. This guarantees that T(0) = T(oe) = 1. Further we take suitable points g l , . . . , g~ E G ~ such that G ~ is generated as an algebraic group by { 9 1 , . . . , 9 ~ } . Take a l , . . . , a s distinct points of C* and fix elements bj E C* with b2 = aj. T h e n the rational m a p S should satisfy: S(aj) = E ( b j ) g j E ( b j ) -1 for j = 1 , . . . , s. T h a t an S with those properties exists follows from the fact t h a t G ~ is, as a variety, isomorphic to an open set of an affine space over C. As we have already seen T is C,~-equivariant and T(0) = T(oe) = 1. Further T(bj) = gj for j = 1 , . . . , s. This implies that G ~ is the smallest algebraic group such t h a t T E G~ I It seems t h a t the technical condition h E N ( G ~ ~ is superfluous. In fact we state the following C o n j e c t u r e : Every linear algebraic group G with G / G ~ cyclic is the difference Galois group of a semi-regular difference equation over C(z).

8.6.

INVERSE PROBLEMS

93

A complete proof is not available at the moment. We consider the example where G ~ is a torus and the conjugation by h. on G ~ is not the identity. It can be seen t h a t h does not lie in N ( G ~ ~

L e m m a 8.12 Suppose that the linear algebraic group G has the properties: G / G ~ is egelic and that G ~ is a torus. Then G is the difference Galois group o f some semi-regular difference equation over C(z). P r o o f : As before, there is an element h E (; of finite order n which maps to a generator of G / G ~ Let G be given as an algebraic subgroup of A u t ( V ) . It suffices to produce an object (V, T) of U~, with Tannakian group G. Let X denote the group of the characters of G ~ . Then h (or C,~) acts also on X. The element T that we have to produce is an injective h o m o m o r p h i s m X -+ {a E C(u,~)* I a(0) = a.(oc) = 1}, which conmmtes with the C,~-actions. Indeed, the condition T(0) = T(oc) = 1 is equivalent to the statement t h a t the image of T, as a h o m o m o r p h i s m , lies in {a E C(u,~)*l a(0) = a(oc) = 1}. The injectivity of T, as a homomorphism, is equivalent to T ~ K(C(u,~)) for any proper algebraic subgroup K of G ~ T h e action of h on the group X makes X into a module over the group ring of C,~ over Z. This group ring can be written as Z[t]/(t '~ - 1). The multiplication by t on X is by definition the action of h on X. Suppose that X can be embedded in a Z[t]/(*" - 1)-module Y, such t h a t Y is a free finitely generated Z-module and such that an in.jective C~-equivariant h o m o m o r p h i s m T : Y --~ {a E C(u~)*l a(0) = a(oc,) = 1} exists. Then the restriction o f T to X has the required properties. The vector space Q @ X is a direct sum of irreducible representations of (7~ over Q. Any such irreducible representation is a direct s u m m a n d of the regular representation Q[t]/(t '~ - 1), where the action of h coincides with the multiplication by t. It follows that X can be considered as a Z [ t ] / ( t " - 1)submodule of (Z[t]/(t n - 1)) N for some N >_ 1. It. suffices therefore to prove the l e m m a for the latter module. Let us first, consider the case X = Z[t]/(t" - 1). Any equivariant h o m o m o r phism Z[t]/(ff ~ - 1) --+ C(u,~)" is given by 1 ~-~ u~,t ~ ~r(w) . . . . . The element ~c, has to satisfy two conditions: (a) u,(0) = u~(>c) = 1 and (b) Wa~

....

= 1 implies that a0 . . . . .

a ~ - i = 0.

The first condition is the equivalent, of T(0) = T(oc) = 1. The second condition is the injectivity of T. un-)~ 1 Choose A1,A2 E C* such t h a t ~ is not a root of unity. T h e n w := ~n-a= satisfies (a) and (b). Thus we have proved the l e m m a for the t o m s with character

94

CHAPTER

8. S E M I - R E G U L A R

DIFFERENCE

EQUATIONS

group Z [ t ] / ( t ~ - 1). Moreover we have produced an element T for this group such that the set of poles and zeroes of T lie in the union of two prescribed orbits in C ~ under multiplication by C,~. For the case ( Z [ t ] / ( t T~ - 1)) N, one defines a homomorphism T by elements

(Wl,... ,WN) of the form: each w j = ~,~-),~(J)~"-x~iJ) The •.(,) are chosen such that the orbits of the 2N elements )~l(j), ~2(j) are distinct. The corresponding T is therefore again injective, C,~-equivariant and T(0) = T(~c.) = l. |

Chapter 9

Mild difference equations 9.1

Asymptotics for mild equations

The following theorem is the tool for the analytic study of mild difference equations. The theorem is a special case of a recent result, of B r a a k s m a and Faber. T h e o r e m 9.1 Suppose that y(z + 1) = A y(z) + a, whelr A and a have coefficients in a finite exteT~sion of k ~ , is a mild equation and let a formal solution S] with coefficieT~ts in 7) be given. There e.rists a meromorphic vector Y~ight, such that:

I. Yright iS defined on a right-domain.

2. Yright has y as asymptotic expansion at ,>c, in a bounded sector with arg(z) E ( ~ - , ~ + e), for some positive ~. 3. ~lright(Z ~- 1) = A Yright(2) 4- a. In general the asymptotic lift Yright is not unique. P r o o f : Let A" denote the "canonical form" of tile difference equation (see Section 7.2). There is an invertible matrix fi with coefficients in 7) such t h a t f l ( z + 1 ) - l A F ( z ) = A c. The equation F ( z + 1 ) = A F ( z ) ( A C ) -1 satisfied by F is a mild equation. We apply now T h e o r e m 4.1 of [16]. We refer to T h e o r e m 11.1 for a translation of that theorem in our notations. The case that concerns us here is part 2. of T h e o r e m 11.1. We note that tile singular directions can only have :t:} as limit directions. Let. 0 < 0 < } be such that (0, 0] does not contain a singular direction. Then the multisum F of F in the direction 0 lives on a sector, containing (0 - ~, 0 + ~). For any direction 9' E (0, 9) one finds the same nmltisum. The conclusion is that F has F as asymptotic expansion on a sector

96

C H A P T E R 9. M I L D D I F F E R E N C E E Q U A T I O N S

( - - ~ , e + ~) for some positive e. Write 9(z) = F ( z ) v ( z ) . Then ~) is a formal solution of the equation v(z + 1) = A ~ v(z). Since this is an equation in canonical form, the vector ii has constant coelficients. Therefore 9(z) := F(z)iJ(z) is a solution of our problem. |

R e m a r k s 9.2 Some notations and comments (1) The homogeneous equation y(z + 1) = A y(z) can have a non-zero solution which has asymptotic expansion 0 in the prescribed sector. Therefore the Yright of the theorem is (in general) not unique. The solution ~]right, given in the proof of the theorem, is the multisum of the formal solution in the direction "0 +''. This choice for Yrigh t is unique. It is functoriat on the category of mild difference modules and commutes with tensor products. We shall see below how this special choice for Yright can be taken as the starting point for the definition of canonical connection matrices. (2) The ring of meromorphic flmctioas f, defined on a right domain and having an asymptotic expansion in Ta on a sector at. oc with opening ( - x2 , 7u+ e ) for some positive e, is denoted by 7~. This is a difference ring containing the field of convergent Laurent series koo. One defines in a similar way a difference ring /2 of meromorphic functions on parts of C. In fact f ( z ) belongs to L; if and only if f ( - z ) belongs to g . In other words, the functions in s are defined on a left domain and have asymptotic expansion in 7) on a sector which is given, with some abuse of notation, by (~, a@ + e). A variant of Theorem 9.1 is: There exists a vector Yl~jt with cooMinates in s such that the asymptotic expansion of yte/t is y and such that Yle/t(z + 1) = A Yleft(z) + a holds.

9.2

C o n n e c t i o n matrices of mild e q u a t i o n s

The idea of the construction of the connection matrix has already been explained. We give here some more details. The mild difference equation y(z + I) = A 9(z) has some symbolic fundamental matrix F. The coefficients of F are combinations of formal Puiseux series and symbols e(9), l with g = c exp(c}(q) - q)(1 + z - l ) ~~ (see L e m m a 7.4 part 3.). On a right domain the Puiseux series are lifted, using Theorem 9.1 and the symbols l and e(g), with g E ~,~ild must be given an interpretation as functions on a right domain. This interpretation has to "cornnmte" with tensor products. Suppose for a moment that we have such an interpretation e(g) ~ f ( g ) , where the f ( g ) are meromorphic functions on right domains. Of course f ( g ) should satisfy

9.2. CONNECTION MATRICES OF MILD EQUATIONS

97

~(f(9)) = gf(g). T h e c o n d i t i o n t h a t the i n t e r p r e t a t i o n " c o m m u t e s with t e n s o r p r o d u c t s " implies t h a t f(gl)f(92) = f ( g l g z ) . Hence f ( 1 ) = 1 and / ( - 1 ) 2 = f ( 1 ) = 1. So f ( - 1 ) = 4-1 and e f ( - l ) = / ( - 1 ) . such an i n t e r p r e t a t i o n is not possible.

T h i s c o n t r a d i c t i o n shows t h a t

T h e g r o u p of the r o o t s of unity, which is d e n o t e d by #oo, is the o b s t r u c t i o n to the c o n s t r u c t i o n of a s u i t a b l e i n t e r p r e t a t i o n of the e(g)'s as functions. We will go as far as possible in giving the e(g) an i n t e r p r e t a t i o n . Some, m a y b e a r b i t r a r y , choices are m a d e in this process. T h e effect of the choices on the c o n s t r u c t i o n of the connection m a p is not essential. Choose a Q - v e c t o r space L0 C R such that. Q @ L0 = R . T h e n every e l e m e n t g C ~,~itd can be w r i t t e n in a unique way as .(1o91 with g0 E #oo and 91 = e~i~~162 - q)(1 + z-l) a~ with a0 C L0 and al E L a n d q = ~ 0 < , < 1 a,z". The subgroup of the elements gl will be denoted by G,~itd,1. We note t h a t the space L can be chosen as L0 | JR. T h e i n t e r p r e t a t i o n s e(g)right, Iright of e(g) and l on a right d o m a i n is now defined as

e(g),.ight = e 2'~i~~ exp(

Z

a~z')e~ 1o~(~)e(go)

and

l,.ight = log(z),

0
where log(z) and z ~ : = e ~l~ are defined with the o r d i n a r y choice log(z) = log(Izl) + i a r g ( z ) and arg(z) C (-Tr, rr]. On a left d o m a i n we make the interp r e t a t i o n e(9)te/t, lle/t is a similar way b u t now using a n o t h e r choice for the l o g a r i t h m of z, n a m e l y l o g ( - z ) + irr. T h e two l o g a r i t h m s coincide on the u p p e r halfptane. On the lower half plane one has ( l o g ( - z ) + i~r) - log(z) = 27ri. T h i s p r o d u c e s a f u n d a m e n t a l m a t r i x F,.i#~t on a right d o m a i n which uses functions and the s y m b o l s e(c) with c C #oo. S i m i l a r l y one finds a s y m b o l i c solution Ft~ft on a left d o m a i n . T h e expression Fright -1 fleft u s e s again the s y m b o l s e(c) with c C #oo a n d functions defined on an u p p e r half plane and a lower half plane. T h u s fi'7:ightFleft -1 is in fact a pair of matrices, say (S~pp~, Sto~er). We will show t h a t S~ppe~ has c o o r d i n a t e s in C({u})[e"~XZe(e-~ix)], where ,~ runs over the set {,~ E QI 0 < A < 1}. As explained m C h a p t e r 8, the ring above is the a l g e b r a i c closure C ~ u } ) of the field C ( { u } ) . We note t h a t the a c t i o n of r on this algebraic closure is trivial. For this reason one can not identify C ( { u } ) with the field of functions O,~_>iC({ul/"~}), since on the l~tter the a c t i o n of r is not the identity! T h e coefficients of St . . . . belong to the field C ( { u -~ }). If A h a p p e n s to have c o o r d i n a t e s in C ( z ) then the two m a p s Supper and St . . . . glue to a "global m a p " S which has c o o r d i n a t e s in K := C(u)[e~'~X~e(e-2~ix)], where ,~ runs over the set {A ql 0 < A < 1}. It is not difficult to see t h a t [~" is a field. T h i s field is the m a x i m a l a~gebraic extension of C ( u ) which is only ramified above the p o i n t s u = 0 a n d u = oc. T h e field K is the union of fields K,~. T h e field Km is the unique extension of degree rn of C ( u ) which is only ramified above u = 0 and

C H A P T E R 9. MILD D I F F E R E N C E E Q U A T I O N S

98

u = oc. The field K,~ has the form C(u,,~) with u{{~ = u. The Galois group of A" over C(u) is isomorphic to the profinite completion Z of Z, i.e. Z is the projective limit of all Z / m Z . A delicate point in the construction is to find the correct canonical choices for Fright and Fl
TO, s 7/+, ~ _ . The first two we have already met in Remarks 9.1. We let "H+ be the ring of meromorphic functions defined on some upper halfplane and having an asymptotic expansion in "P for a sector with arg(z) E (~, @ + e), and e > 0. Similarly "H_ denotes the ring of meromorphic functions defined on some lower halfplane 7r 7r having an asymptotic expansion in 7) for a sector with arg(z) E ( g, .7 + e), and e > 0. The difference ring 7)Mild is the ring introduced in L e m m a 7.4, 7)Mild = 7)[{e(g)}ge~,,~,l], where Gmild denotes the subgroup of c.j consisting of the elements {e e * V ( * ( q ) - q)(t +

C C ' , a0 e L, q =

a.z"}. 0<#
This is a difference ring of "formal expressions". The next. difference ring 7r : = "~[e(.f])right, lright] uses functions and the symbols e(c) with c E #oo. The definitions of s ~ + M i l d , J i _ M i l d are similar. The choice of the logarithm is not important for the last two rings. Let M be a mild difference module over C(z) (or over C ( { z - 1 } ) ) . One defines spaces of solutions of M as follows:

wo(M) = k e r ( ~ - 1 , P M i l d O M ) wright(M) = k e r ( ~ - 1,7~Mild | M ) wz~jt(M) = ker(Op- 1 , s

M)

w~vw,.(M) = ker(O~ - 1,Tl+Mildr

M)

wt . . . . (M) = ker(O~ - 1 , T l _ M i l d O M ) The tensor products are taken over C (z) if M is defined over that field, otherwise they are taken over C ( { z - 1 } ) . The comparison of these spaces is the main tool in the construction of the connection m a p and the resulting theorems.

9.2.

CONNECTION

Lemma

MATRICES

OF MILD EQUATIONS

99

9.3 In the following u denotes e u~i~.

1. The set of r

elements of T t + M i t d is equal to the field C ( { u } ) .

2. The set of O-invariant elements of T i _ M i l d is C ( { u - ~ } ) . 3. The sets of r the field I( = U,~t(m.

elements of 7~Mild and s

are contained in

P r o o f : (1) Let. f = ~ c r 1 6 2 fee(c) be a r element. The coefficients fc are taken in 7t+[e(g),.ight,l,.ight]9~Gm,,d.~. The equation O(f~)c = fc has the consequence fc = e - 2 ~ i ~ h ( u ) , where k E Q is taken such that 0 _< A < 1 and e 2'~ix = c. The function h is defined on 0 < ]u] < e for some positive e. The functions in 7t+[e(g)~igm,l~ight]g~.,~,,d.~ can be bounded on a sector ({, 2 + e) by c~e ~I~1 for some positive constants c~,e2. From this one deduces that h E C ({u}). This proves the inclusion of the set of the r elements of 7-l+Mild into C({u}). The other inclusion is shown by constructing special elements. We recall that a choice of a Q-linear subspace L0 C R is made such that L0 9 Q = R. Clearly L0 and rn + L0, with any rn E Z, are dense subsets of R. Thus there exists a b = b0 + m with 0 < b < 1, b0 G L0. The element 'tte-2rribze(e2rrib)right is r and ue -2~i~ has a s y m p t o t i c expansion 0. Thus u e -2rcibz belongs to 7-l+ and Ue-2rribZe(e2rrib)right is a r invariant element of 7-t+Mild. From the definition of the e(g),qght it follows that U e - 2 r r i b z e ( e 2 7 r i b ) r i g h t = U- m + l . Clearly, also C{u} and its integral closure C{u} = U,~>~e{u,,~} belong to the set of r elements of 7 i + M i l d . Combining this one obtains the other inclusion. This proves (1). T h e proof of (2) is similar. (3) A r element f of g . M i l d is defined on the whole complex plane. The restrictions to an upper and a lower half plane lie in the fields C ( { u } ) and C ( { u - 1 } ) . From this (3) at once, We suspect in fact that the r elements of'PvMild and s are constants. | We note that w0(M) is a vector space over C and that the canonical m a p 7) M ild O c Coo(M) ---+P M iId | ) M is a bijection. L e m m a 9.4 Multisummation in the di,vction 0 + induces a C-linear map ~oo( M ) ---+Wrigm ( M ) . A basis of coo ( M ) over C is mapped to a basis of co,.ight ( M ) over the ring of O-invariant elements of'l~Mild. P r o o f : Let y E w0(M). The element y has uniquely the form

=

~ g ~ , r , ttd, 7Z~O

~(g,n)e(g)l '~ where ~ ( g , . ) C ~.

100

C H A P T E R 9. MILD D I F F E R E N C E E Q U A T I O N S

The equation ~(y) = y leads to a set. of equations for the y(g, n), namely E

r

n))(l + log(1 + z - ; ) ) '~ = g -~ E

Y(9, n) l'~ for each 9.

These equations are mild and according to Theorem 9.1 there are lifts Y(g,n),-ight E ~ of the ) ( 9 , n ) such that the {y(g,n)right} satisfy the same equations. In general the lifts are not unique! The Y(9, ~lT,)right that we take are defined by multisummation in the direction 0 +. The element Yright := ~'-~.g,,~Y(g, n)right e(g)rightlriq~zt, with the interpretation and lright a s explained before, is 4P-invariant and consequently lies in This defines the map in the lemma. By our construction the module M trivializes over T~[{e(9)}~,~,,,~,l] and hence also over T~Mild. A basis of w0(M) is therefore mapped to a basis of~v,.i~t~t(M) over the ring of 0-invariant elements of ~vM ild. |

e(g)right COright(M). of

There is also a natural map wo(M) --+ wl
--4, w~pp~,.(M).

Combining with the previous maps one finds a "left" and a "right" isomorphism

We define an upper connection ,nap SM, upper ~ GI(C({'u}) @c w0(M)) by first taking the "left" map and then the inverse of the "right" map. The definition of 5'M. to~," is similar. This map has coefficients in C({u-*}). T h e o r e m 9.5 Let M be a mild difference module over C(z). (1) The upper and lower connection maps of M are the restrictions of an automorphism S M of [(rn @ wo(M) with some m >_ 1 and Km the unique field esctension of C(u) of degree rn which is only "ramified above the points u = 0 and U~

00.

(2) If M is very mild, then SM has coordinates in the field C(a). Moreover S t , and S -M1 are regular (i.e. defined) at the points u = 0 and u = co. (3) The difference Galois group of M over C ( : ) is the smallest algebraic subgroup G of Gl(wo(M)) ~- G l ( n , C ) such that G contains the formal difference Galois group and SM C G(Km). P r o o f : (1) Since the equation is defined over C(z), the coordinates of the upper and the lower connection maps are combinations of functions meromorphic on all of C and symbols e((~) for some roots of unity ~. Moreover the two connection

9.2. CONNECTION MATRICES OF MILD EQUATIONS

101

maps coincide and we can introduce the symbol SM for both maps, Lemma 9.3 implies that that the coordinate of SM are in f~'m = C(um), for some rn _> 1 frt and with u~ = u. (2) For very mild equations one can make the following variation on the construction of the connection maps: The difference rings 7)Mild, T~Mild,... are replaced by subrings 7)VeryMild, 7~VeryMild .... which are given as

7)VeryMild = P[{e(g) }ge~ . . . . . ,,~, l], T~VeryMild =/'~[{e (g)right }gE6. . . . . ,,d, lright], etc. The spaces W~ight(M),wlest(M),... are now defined with respect to those subrings 7~VeryMild, s For the "very mild case" one has the following version of Lemma 9.3: 1. The set of the r

elements of 7t+ VeryMild is equal to the ring

. The set of the r

elements of Jt_ VeryMild is equal to the ring

The main point is that an element h of ~ + VeryMild can be bounded on a 71" 71(7, g + e) by cle c21zl" with positive constants c1,e2 and 0 < # < 1. If h is moreover r then this implies that h E C{u}. sector

This leads to isomorphisms

c{u} r C{u -1 } |

~o(M) -~ "~upper(M), wo(M) -+ Wlowe,-(M).

Hence SM, upper and SM, lower are invertible maps on the spaces C { u } | and C{u -1 } | w0(M). Hence SM = SM, upper = SM, lo~er and its inverse have coordinates in the subring of C (u) consisting of the rational functions which are defined at 0 and at oo. (3) The map SM depends in a functorial way on M. Its construction commutes with "all constructions of linear algebra", in particular with tensor products. It follows that the difference Galois group (7 must satisfy SM E G(['(m) (and of course G contains the formal difference Galois group). On the other hand, let Z denote the smallest algebraic subgroup of Gl(wo(M)) containing the formal difference G alois group and satisfying SM E Z(Km). Then the set of Z-invariant elements of ~00(M) is easily seen to be ker(d~- 1, M). From this one can conclude that Z = G. l

C H A P T E R 9. MILD D I F F E R E N C E EQUATIONS'

102

A similar proof can be worked out for an analytic mild difference module, i.e. a mild difference module over ko~. C o r o l l a r y 9.6 Let M be a mild difference module over koa = C({z-~}). The upper and lower connection matrices have coordinates in the fields C({u}) and C ( { u - ~ } ) . The difference Galois group of M is the smallest algebrmc subgroup G of Gl(~o(M)) such that:

1. GM, f . . . . at C G.

2. sM, S. SM, I. . . .

~ GCCC{u-1})) +

C o r o l l a r y 9.7 Let M be a mild difference module over C(z) then M and ko~ | M have the same difference Galois group. Proof." The difference Galois group G of koo @ M is certainly contained in the one of M. From S M , upper E G ( C ( { u } ) ) it follows that S M ~ G(K) since S M = SM, upper. According to 9.5, G is also the difference Galois group of M.

| We take a small break in order to point out a consequence of Corollary 9.7. A result of Birkhoff states that every differential module M over koo comes from a differential module N over C(z) (i.e. M is isomorphic to koo | N). It is somewhat of a surprise that this does not hold for difference equations. We give two examples. E x a m p l e 9.8 The difference equation y(z + 1) = (1 + z - 1 ) l / m y ( z ) is not equivalent over koo with a difference equation over C(z). To see this consider the matrix difference equation

y ( z + 1) =

e 2~ri/n

0

0

(l + z_l)t/m

) y(Z)

The formal fundamental matrix of the equation is

e(e rci/n) 0

0 e((l+z-1)l/m))

) "

Its difference Galois group over koo (and also over ]%o) is Z / n Z • Z / m Z . If a difference equation over C(z) is equivalent with our example then this equation is also mild and has therefore the same difference Galois group. However for a difference Galois group G over C(z) one knows that G / G ~ is cyclic. This contradiction proves that the matrix difference equation is not equivalent to a

9.2. CONNECTION MATRICES OF MILD EQUATIONS

103

matrix equation over C(z). Since the upper part of the equation is already defined over C(z) it follows that the lower part is not equivalent over koo with an equation over C(z)

E x a m p l e 9.9 The difference equation y(z + 1) = (1 + z-2)l/my(z) over koo does not come from an equation over C(z). To see this note t h a t the equation is regular and has an upper and a lower connection matrix, called Supper and .sl...... For the equation v(z+ 1) = (1 +z-2)v(z), which is the ruth tensor power of the first equation, we will show later in 10.2 that the connection matrix is ~)(u-e-2") 95' = ( u - - e 2(u_l)2 Hence .Supp~ = 8 1 / m = 1 + *u + *u 2 + ... E C { u } and a similar expression for Slo~,er. If the equation of the example came from an equation over C(z) then Supper is the germ at u = 0 of a rational function in u. However s is not an ruth power in C(u).

C o r o l l a r y 9.10 Let M be a regular singular difference module over C(z).

1. The connection matrix SM has coordinates in C(u). Further SM(O) = 1 and S M (00) is the formal monodromg map of M. 2. The difference Galois group of M is the smallest algebraic subgroup G of Gl(wo(M)) such that SM E G ( C ( u ) ) . In particular, the difference Galois group of M is connected. P r o o f : 1. According to T h e o r e m 9.5, the coordinates of SM are in some K,~ = C ( u , , ) . The number m _> 1 corresponds to the occurrence of the ruth roots of unity in the formal classification of the equation. In the regular singular case m=l. Let y C wo(M) have the form

y=

?)(a, n)e((1 + g--1)a)ln.

E

a E C / Z , n>O

T h e term C / Z indicates {a E C I 0 _< Re(a) < 1}. Then one has, with the obvious notations, Yright "~

E

Y(a'n)r~g hteal~176

at=C/Z, n > O d

y(a, ,,)~ejte a0~

Y l e f t -~

z) + i~) ~

aEC/Z, n>O

On the upper hMf plane log(z) = l o g ( - z ) + i r r and the y(a, n)right and y(a, n)le.ft have the same a s y m p t o t i c expansion. This implies SM(O) = 1. On the lower

C H A P T E R 9. MILD D I F F E R E N C E E Q U A T I O N S

104

half plane l o g ( - z ) + irr = log(z) + 27ri and tile fornmla tbr YleJt can be written there as Ylelt

=

~

~,(,~ ~ ,

. .2rria.alog(z) ~,~, .o,~:t~ ~ (log(z) + 2~-i) ".

aeC/Z, n_>0

From this the statement about SM(OC) follows. 2. The formal difference Galois group is easily seen to be generated by the formal m o n o d r o m y map. From T h e o r e m 9.5 and SM(OO)= the formal m o n o d r o m y group, the first, statement follows. The group G is connected since SM(O) = 1.

I 9.11 For an analytic regular singular difference module M, i.e. M is defined over koo = C ( { z - Z } ) , Corollarg 9.10 has an obvzous analogue:

Remark

1. SM, ~ppe~(O) = 1 and SM, t . . . . (oo) is the formal monodromy map. 2. The difference Galois group is the smallest algebraic group G such that SM, ~p~ ~ G ( c { W ) a n d sM, t . . . . . e O'(C{a-~}). Let d be a complex number and y(z + 1) = A(z)y(z) a difference equation. The shift by d of this equation is the equation y(z + 1) = A(z + d)y(z). Note t h a t y(z) satisfies y(z + 1) = A(z)y(z) if and only if ~(z) = y(z + d) satisfies 9(z + 1) = A(z + d)~(z) Let M denote the module corresponding to the matrix A(z), then raM will denote the module corresponding to the matrix A(z + d). 9.12 Let S = S(u,~) denote the connection matrix (total, upper or lower) of the mild difference module M, then S(e2~id/mu,~) is the connection matrix of the shifted module raM. Lemma

P r o o f i We will quickly verify that the shift re can be defined on all the objects t h a t are needed for the construction of the connection matrix of a mild module and t h a t re c o m m u t e s with the construction of the connection matrix. Indeed, rd acts on 7) by re(z A) := ( l + d z - 1 ) X z A. Further re(l) := l + l o g ( l + d z - 1 ) . For g = e exp(O(q) - q)(1 + z - l ) a~ E Umild one defines

l+(dH-1)/z r(g) ---- (1 -t- d/z)(1 + 1/z))a~

- q) - (vd(q) -- q))g.

We note t h a t g is multiplied under the action of ra by a convergent Puiseux series since ra(q) - q is a Puiseux series in positive powers of z -1 The action of re on 7)Mild commutes with the action of r One defines the action of re on 77~,s in the natural way, i.e. (ref)(z) = f ( z + d). Next one cai1 verify that the natural action of re on the functions e(#)right , e(#)left, lright, lleft is compatible with the re action on the symbols e(g), l for g E ~rnild. This defines an action of re on TiMild et cetera, still c o m n m t i n g with r Finally, the unicity of the m u l t i s u m m a t i o n proves t h a t Yright (Z + d) coincides with the lift of the the shifted formal solution of the shifted difference equation. I

9,3. TAME DIFFERENTIAL MODULES

9.3

Tame

differential

105

modules

In order to obtain nice formulas for connection matrices we extend the construction of the connection matrix to tame difference modules. A difference module M over koo (or over C(z) ) will be called tame (over ko~ ) if M can be written as a finite direct sum

M = E

k~e(z~) | M),, where

;~6Z

each Mx is a mild difference module because the modules Mx are supposed One can extend this notion as follows: koz(z 1/rn) ofkc~ if koz(z 1/rn) | M can

k~

E

over k ~ . This decomposition is unique to be mild. M is tame over the finite field extension be written as a finite direct sum

]r

where

A61/mZ

each M), is a mild difference module over The decomposition is again unique.

koo(zll~).

The module is tame (over k~(z ~/'~) ) if and only if M can be represented by a matrix difference equation y(z + 1) = A y(z) (over k~(z I/'~) ) such t h a t A is a direct sum of blocks. Each block has the form za(Bo + B1/,~z -]/'~ + ...) with A E Q and B0 an invertible matrix. For a t a m e module M we want to define the connection matrix. T h e space of formal solutions ~ 0 ( M ) is the kernel of 9 - 1 acting on P[{e(g)}ge~, l] | M. We have to make a decision how to lift the symbol e(z) to a elements e(z)right and e(z)le/t. In the literature one finds more than one choice. Here we will take the same choice as G.D. Birkhoff in [9], namely:

e(Z)righ t : r ( z ) a n d e(Z)lef t =

-2~iei'Zr(1 -

z) - 1 .

Reasons for this choice are the following. The F-function has no zeros on C \ ( - o c , 0] and is then the exponential of a unique holomorphic F(z) on the same domain with F(1) = 0. Thus we can define e(zX)~ight as e xF(z). Similarly ei~z F ( 1 - z)-1 has no zeros on C \ [1, oo) and is the exponential of a holomorphic function G on the same domain with G(0) = 0. Then e(Z)')lejt will be e xG(z). The constant in the expression for e(z)l~/t are chosen such t h a t the "connection matrix" of the equation y(z + 1) = zy(z) has a simple form. This connection

matrix is e(z);/ght

(z)

1t

=

1 -- u

T h e natural definition of the connection matrix S of a tame difference module M = }--~ k~e(z ~) | Mx is the direct sum of the connection matrices of the

C H A P T E R 9. MILD DIFFERENCE EQUATIONS

106

pieces. The connection matrix of kooe(z x) @ M~ is the product of the connection matrix of Mx with the constant (1 - u) x. The meaning of (1 - u) x is defined by the choices of F and G above. In particular (1 - u) x has value 1 for u = 0. Finally we note that connection matrices for tame modules respect tensor products.

9.4

I n v e r s e p r o b l e m s for m i l d e q u a t i o n s

In the first equations. equations. koo with a

part of this section we will study equivalence classes of mild difference This theme returns, in a more complicated setting, for wild difference The last part gives a construction of a mild difference equation over prescribed difference Galois group.

Fix a mild difference equation y(z + 1) = 5; y(z) over the algebraic closure of ko~. We consider the set of all difference equations y(z + 1) = A y(z) which are formally equivalent, i.e. over the field 7), to y(z + 1) = S y(z). As we know, the formal i s o m o r p h i s m / ~ with F ( z + 1 ) - l A / ? ( z ) = S can be lifted to Fright on an "extended right half plane", i.e. on a sector of the form - ~ 2 < arg(z) < ~ + 5 for some 5 > 0. There is also a lift Ft~.rt on an "extended left half plane", i.e. on a sector of the form

{z ~ Cl

7r

z = Izle i,r Izl > R, ~- <

r

3~r

< ~- + ,s}.

The positive n u m b e r ~ depends only on S. The matrix T

:=

--1 fleftfright has the properties:

1. T is defined on the union of two sectors y < arg(z) < y~ + 5 and --~ < arg(z) < - ~~ + 5. 2. T is asymptotically the identity on this union of the two sectors, i.e. this property holds on any proper subsector defined by <_ signs.

3. T(z + 1)-1ST(z) = S. We will call a matrix T with the properties (1),(2) and (3) above a connection cocyele. To the pair (A,/#) we associate the connection cocycle F(~)tF~ight. This connection cocycle is not unique, since the lifts Fright and Flair are not unique. A n o t h e r choice of those two matrices changes T into UlejtTUright where Uright is asymptotically the identity on the extended right half plane and Uright(Z + 1)-lsUright(Z) : S. The other matrix U~]t has similar properties. We call T and Uze]tTUright with the stated properties equivalent. Let H 1(S) denote the set of equivalence classes of the matrices T.

9.4. I N V E R S E P R O B L E M S FOR MILD EQUATIONS

107

On the set of pairs (A, fi') satisfying/>(z + 1)-lA/~(z) = S we introduce also an equivalence relation. The pairs (Ai,/~i), i = 1, 2 are called equivalent if there is an invertible meromorphic matrix C, defined over the algebraic closure of k~, such that C(z + 1)-IA1C(z) = A2 and F1 = c F s . Equivalent pairs produce equivalent connection cocycles.

T h e o r e m 9.13 The map from the set of equivalent classes of pairs (A, iv), satisfying [Z(z + 1)-lA/~(z) = S, to H i ( S ) is bijective. P r o o f : The verification of the injectivity of the map is a straightforward matter. We will prove now that the map is surjective. Let T be a connection cocycle. The Theorem of Malgrange-Sibuya (see section 8.4) asserts that there is an invertible formal matrix F (with coefficients in 7)) and two asymptotic lifts F1, F2 of/~ defined on sectors ( - 2 ' ~ + 5) and (~, ~ + 5) (with some abuse of notation) such that F~IFs = T. The two invertible meromorphic matrices Ai = Fi(z + 1)SFi(z) -1, i = 1,2 are defined on the two sectors above. On the intersection of the two sectors one has A1 = As. Hence the pair A1, As glues to an invertible meromorphic matrix A with coefficients in the algebraic closure of ko~. The connection cocycle is obviously the image of the pair (A,/?). | C o r o l l a r y 9.14 Suppose, in addition to the theorem, that the coefficients of S are in C(z) and suppose that the connection cocycle T extends to an invertible meromorphic matrix on all of C. Then the connection cocycle is the image of a pair" ( A , F ) such that the coefficients of A are in C(z). Proof.' We use the notation of the proof of the last theorem. Let the pair F1, F2 satisfy F~-IF2 = T. Then clearly F1 and F~ are invertible meromorphic matrices on all of C. The same thing holds for A = Fl(Z + 1)SFI(z) -~ = Fs(z + 1)SFs(z) -1. Since A is also invertible meromorphic at oc one has that the entries of A are in C(z). | R e m a r k 9.15

The multisummation theorem associates to a pair (A, [7) w i t h / ~ ( z + l ) - l A / ? = S unique asymptotic lifts Fright and Ft~ft on an extended right half plane and an extended left half plane. Thus multisummation provides a connection matrix -1 T = FleftFright which is a representative of the equivalence class of the connection cocycle in H i ( S ) of (A,/~). T h e o r e m 9.16 Every linear algebraic group G which has a finite commutative subgroup I~ with at most two generators and such that K ~ G / G ~ is surjective is the difference Galois group of a mild difference equation y(z + 1) = A y(z) over the field k ~ .

108

CHAPTER 9. MILD DIFFERENCE EQUATIONS

P r o o f : In principle we will use Corollary 9.6 in order to obtain the required difference equation. However this corollary is not very constructive and we will in fact work with a subclass Ad,~,~ of the mild difference equations for which the connection matrices (with a slightly different definition) and the formal monodromy can be prescribed. For this subclass we will use the method of the proof of Theorem 8.10. Fix two integers n , m > 1. A mild difference module M over kcc belongs to this subclass if ]%o[e(e--22'~i/'~),e((1 + z-1)-1/~)] | M is a trivial difference module. In other words, there exists a fundamental matrix for M with coordinates in the difference ring ]%o[e(e->~i/"), (e((1 + z-1)-1/'~)]. The group of the automorphisms (commuting with r and ]r of this difference ring is denoted by H. It is the product of two cyclic groups of order m and n. The two generators o"1, o'2 are given by

ohe(e -2film) = e2'~il~e(e-2~il~); crle((1 + z-i) -1In) = e((1 + z-1)-lln), o2e(e -2rri/m) -~ e(e-2rri/m); 0"2e((1 -l- z - l ) -l/n) : e2rci/ne((l'q- Z-1)-l/n). This group will act on the constructions that we will make. We introduce difference rings 7~',s 7/'_ as the subrings of 7~, Z:, 7/+, 7/_ having their asymptotic expansion in ]%o. The spaces w0(M), ~o,ight (M), ... are now defined as

ker(~ - 1, ]%o[e(e-2"i/m), e((1 -I- Z-1)-l/n)] @ M), kgr(~ -- 1,~'~'[e(e-2"i/rn),e((1 -~ Z-1)-l/n)]

@

M),

etc. We note that we keep the symbol e((1 + z-l) -l/n) and will not replace it by, say e((1 + z-1)-V'~)right. On each of the space wao(M),wright(M),... there is an action of H. As in the general method for mild equations we obtain two connection maps, again denoted by SM, ~,pp~r,SM, tow~. They are not the same as before since we have not given the symbol e((1 + z-l) -1/'~) an interpretation. The two connection maps are H-equivariant. Moreover SM,,,pp~,(O) = 1, SM, to~oe~(oo) = 1. The upper connection map is an invertible m a p with coordinates in the ring of the r -invariant elements of 7/~_[e (e- 27ri/m), e ((1 + z - 1) - 1/n )]. We will need this ring explicitly. A r of the form

element is a sum of expressions

~ -2rria/m )e((1 + z-l) -bl'~) with 0 _< a < m , 0 _< b < n e2rciz~ znha,b(u)e(e and that

ha,b(U) a meromorphic function of u is some set 0 < lul < e. The condition / X e2rciz~~ Z b~ .na,b(U) has an asymptotic expansion in ]ca implies that h a , b lies

9.4. INVERSE PROBLEMS FOR MILD EQUATIONS

109

either in C{u} or u C { u } . Put F1 = e2~iZ/~e(e -2~i/~) (this was called um earlier) a n d / ; 2 = zU~e((1 + z - 1 ) - ~ / ~ ) . We note that F~ = 1. Then the ring of the r elements is a subring of C{u}[F~, F2]. It is in fact the subring C{u}[F1, uF2, uF~,.., uF~-l]. This is an analytic local ring of dimension one over C. A similar calculation can be done for the determination of the ring of r invariant elements (7/'_[e(e-2~i/"~), e((1 + z - 1 ) - l / ' ~ ) ] ) r For notational convenience we denote the two rings of r by Aupper and Alow~. We have now associated to M in the subclass A/lm,,~ the following object (V, Tu,TI) = (wo(M),SM, u p p e r , S M ' l . . . . ) with: (a) V a finite dimensional vector space over C with an H-action. (b) T~ is an H-equivariant a u t o m o r p h i s m of Auppe r @ V such t h a t T~ (0) = 1. (c) Similar s t a t e m e n t for Tl. As in the proof of T h e o r e m 8.8, one can show that the subclass Jt4m,n is equivalent to the category of all triples (V,T~,Tl) as above. T h e T a n n a k i a n group of a triple is the smallest group G which contains the action of H on V and satisfies Tu E G(A~vw~),T~ C G(A~. . . . ). The connected c o m p o n e n t G ~ is in fact the smallest algebraic group with T~ E G~ Tt E G~ The m a p H -+ G -+ G/G ~ is surjective. Let now G E GL(V) satisfying the conditions of the t h e o r e m be given. Choose suitable m , n and a surjective H --+ K. Then we should produce a triple (V,T~,T~) such that G ~ is the smallest algebraic group with T~ C G~ C G~ We start by taking Tt = 1. Further T~ corresponds to an H-equivariant h o m o m o r p h i s m of C-algebras ~b' : O(G ~ -+ Auppe r such that the preimage of the m a x i m a l ideal of Aupper is the m a x i m a l ideal of the point 1 E G ~ It suffices to produce a ~b' which is also injective, since in that case T~ (~ Z(Aupper) for any proper closed subset of G ~ Let Oao,1 denote the local ring of G ~ at 1 E G ~ Let d be the dimension of G ~ There is a basis tl ,t d of the m a x i m a l ideal of OGo,1 such t h a t each tj is an eigenvector for the action of H. In other words, there are 0 < aj ( m and 0 < b j < n such t h a t a l t j = e 2 ~ i a a / r n t j ; ~ 2 t j ~-- e2~ibj/ntj. T h e analytic local ring of G ~ at 1 is denoted by O Ga~~ ,1 " This ring has also tl , 9 .. ta as generators for its m a x i m a l ideal and so 0 Go,1 a'~ = C{t 1.... ,td}. An H-equivariant h o m o m o r p h i s m of local analytic rings ~ : O~ol -+ Aupv~ C C{u}[F1, F2] is now described by: each tj is m a p p e d to an element of U~FF'Fb'C{u} ( w i t h e equal to 0 or 1). It is clear t h a t one can choose the images "~(tj) such that they are algebraically . . . .

independent over C. This implies that ~b' : O(G ~ --+ 0 ~ , 1 ~ A,pv~ is injective.

l

110

CHAPTER 9. MILD DIFFERENCE EQUATIONS

R e m a r k 9.17 The condition in Theorem 9.16, imposed on the group G, is also necessary for the group to be a difference Galois group of any difference equation over ko~. This will be proved in Proposition 10.11.2.

Chapter 10

Examples of equations and Galois groups 10.1

Calculating connection matrices

For a regular difference module M over C(z) or koo we will give more or less explicit formulas for the connection matrices. A regular module has a presentation in matrix form y(z + 1) = A y(z) wit, h A = 1 + A2z -2 4The formal fundamental matrix F has the tbrm 1 4- F l z -1 + .... The right and left lifts of F is denoted by F,.sght and F ~ / t . The connection matrix is F ~ h t F t ~ / t and, in general, is defined for z E C with lira(z)[ > > 0. Its restriction to {z C C[ Ira(z) > > O} is the matrix of SM, upper o l 1 a suitable basis of w0(M). Similarly for SM, t o , ~ . If M is defined over C(z) then A = 1 + A2z -2 + ... is chosen with coefficients in C(z). In that case F ~ h t F z ~ j t is defined on all of C and is equal to the matrix of SM with respect to a suitable basis of coo(M). .

.

.

.

P r o p o s i t i o n 10.1 Let M be a regular differe~ee module over koo or over C(z). Let a matrix equation y(z + 1) = A y(z), representin 9 M , be chosen such that A = 1 + A . , z - ~ + A 3 z - 3 + . . . and A has rational coefficients if M is defined over C(z). The formal fundamental matrix F has the form 1 + F l z -1 + ... and its lifts F,-ight and Flirt are equal to lira A ( z ) - l A ( z lira A ( : -

1)A(z

+

1)-l...A(z - 2)...

+ n-

1) -1 and

A ( z - ,, + 1).

The connection matrix ~.ightFleft 1 is equal to

lira A ( z 4- ~ - 1 ) . . . A ( z ) A ( z - 1 ) . . . A ( z - n + 1). ~ --+ C ~

C H A P T E R 10. E X A M P L E S OF E Q U A T I O N S A N D G R O U P S

112

P r o o f : F r o m F~ight(z + 1) = A(z)F~,ght(z) it follows t h a t

Fright(Z) = A ( z ) - I A ( z + 1) - 1 " " .A(z + n - 1)-lFright(z + n). It is easily seen t h a t l i m , ~ o o A(z) - ~ . . . A(z + n - 1) -1 converges (locally) uniformly on a r i g h t - d o m a i n . F u r t h e r lim~-~o~ F~.ight(Z + n) converges (locally) u n i f o r m l y to 1. This proves the first s t a t e m e n t . T h e second s t a t e m e n t can be o b t a i n e d by replacing z by - z . T h e t h i r d assertion is now obvious. | Example

10.2

The equation y(z + 1) = a(z)y(z) with a(z) E C ( z ) * .

Every order one e q u a t i o n y(z + 1) = ay(z) over C ( z ) is t a m e a n d defines a connection m a t r i x which is an element in C(u)*. We will d e n o t e the i n d u c e d m a p C(z)* --+ C(u)* by C. Since the connection m a t r i x r e s p e c t s tensor p r o d u c t s , C is a h o m o m o r p h i s m of groups. Let C(z)~it d d e n o t e the elements of C(z)* which have value 5s 0, oo at oc. Every element of C(z)* has uniquely the form z'~m with n E Z and m mild. By definition C(z'~m) = (1 - u)'~C(m). We want to prove t h a t (Cf)(e2~idu) = C(rdf)(u). For mild elements this has been proved in L e m m a 9.12. Hence we have only to verify this p r o p e r t y for z. T h i s a m o u n t s to proving t h a t 1 -- C 2 r r i d u

Lemma

10.3

C(1 -}- dz - 1 ) -

1--u

P r o o f : W r i t e (1 + d/z) = (1 + z-1)db. T h e e l e m e n t b E C { z - t } is regular. T h e s y m b o l i c solution I of the e q u a t i o n y(z + 1) = (1 + d/z)y(z) has lifts fright = e al~ l-L~=0 b(z + n) -1 and f,r -- e - ~ 0 ~ l-I~_-a b(z - n) -1. T h e n fright has an a n a l y t i c c o n t i n u a t i o n on all of C. Using the f o r m u l a for f~ight and the e q u a t i o n f~ight(z + 1) = (1 -- az-X)fright(Z) one finds t h a t fright has poles of order 1 in a s u b s e t of a + Z and has zeros of order 1 in a s u b s e t of Z. For flr one finds poles of order 1 in a subset of Z and zeros of order 1 in -1 a s u b s e t of a + Z. T h e connection m a t r i x s = frightfle]t has as function of u a pole of order 1 f o r u = 1 and a zero of o r d e r 1 for u = e 1 --e2~idt~ s(0) = 1 leads then to the required f o r m u l a s = 1-u

2~id. T h e c o n d i t i o n |

Now t h a t we know t h a t C has the correct b e h a v i o r with respect to shifts a n d is a h o m o m o r p h i s m one can easily prove the f o r m u l a

cz'~ l-I(z - cj) '~j ~-~ (1 - u) ~ H ( 1 - e-2"iC'u) '~j, where c, cj E C*. J j In o t h e r terms, the c o n n e c t i o n m a t r i x of a : = cz '~ 1-Ij(z - cj) '~, with divisor n[0] - n[oa] + ~ nj[ej], is the unique element s C C(u)* with s(0) = 1 and with divisor n[1] - n[oo] + ~ nj[e2~i%].

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113

R e m a r k s 10.4 (1) First we make some c o m m e n t s on the proof that the connection m a t r i x of (1 + d/z) is s ~_~ The evaluation s(oc) = e 2~id is (as it should be) the formal monodromy. Further the connection matrix s E C(u)* has the form s~pp~ = 1-I~___oo b(z + n) and S , o ~ = e 2~d I ] ~ = - ~ b(z + n). This shows t h a t the p r o d u c t 1-I,~~176 b(z + n) converges on the upper halfplane to i - ~J---u %

and

on the lower halfplane to e -2rrid l-e-2"=u (2) According to 2.31 the difference Galois group of the equation y(z + 1) = ay(z) with a E C(z)* over C(z) is equal to the cyclic group of order m if and only if a has the form ; / ( ~ + i ) for some primitive ruth root of unity (" and some f E C(z). ", f(z) One can reformulate this as follows:

The equation has a difference Galois group of order m >_ 1, if and only if." (*) a(oc) is a primitive ruth root of unity and (**) The restriction of the divisor of a to any Z-orbit in C (i.e. a set of the form c + Z C C) has degree O.

T h e formula for the connection matrix s of the equation y(z + 1) = ay(z) with a = cz~I~j(z - cj)~J with c, cj E C* shows that s = 1 if and only if the divisor ~ nj[e2~icJ] is trivial and n = 0. This is precisely the same as a(oo) E C* and the condition (**) above (3) Every element t E C(u)* with t(0) = t(oc) = 1 is the image under C of a regular element a E C(z)*, i.e. with a(oo) = 1. This is a special case of Corollary 8.6. (4) We have seen t h a t the connection matrix of the equation y(z + 1) = zay(z), which was produced in a formal way, "commutes" with the shift operators rd. This implies that the connection m a p of any tame differential module also "commutes" with the shifts ra.

Example

10.5 The equation y(z + 1) - y(z) = a with a E C ( z ) .

a \ with a E C(z). The equation y ( z + l ) = A y(z) is very mild. / Indeed, put B : = ( 01 b1 )witt3. a suitable b E C[z]. Then Put A =

01 ) 1

B(z + 1 ) - t A ( z ) B ( z ) is equal to ( 1 0

6I ) w i t h a E C ( z )

suchthat~(oc)=0"

C H A P T E R 10. E X A M P L E S OF E Q U A T I O N S A N D G R O U P S

114

The symbolic fundamental matrix F has the form

0

()wit.

f C C ( ( z - ~ ) ) + C l satisfying f ( z + 1) - f ( z ) = a. The connection matrix reads ( 1 b "~ with b E C ( u ) s u c h that b ( 0 ) = 0. The connection matrix defines < 0 1 ] ' a m a p C(z) --+ C(u) which will be denoted by s We will derive an explicit formula C-linear m a p s The key for this is the formula:

L e m m a 10.6

1 2 ~ /~

s

-1) -

1-u

P r o o f ; Write z -1 = b + log(1 + z - i ) . Then b r z - 2 C { z - X } , T h e symbolic solution f is equal to l + B with B ~ z - ~ e [ [ z - 1 ] ] . Then f~ght = log(z) ~,~=o b(z + n) and fz~ft = l o g ( - z ) + irr + ~,~=~ b(z - n). The function f~ight has an analytic continuation to C and has poles of order 1 in a subset of the negative integers, Similarly fl~It has poles of order 1 in a subset of the positive integers. s -1) is the difference of those two functions and has a pole of order 1 foru= landazeroforu=0. Hences ,-~, for some constant c. The expression f l ~ l t - fright is on the lower halfplane equal to 27ri+ ~,~___~ b(z + n). For z --+ - i o c the infinite sum tends to zero. This proves that c = -27ri. I It can be seen that s has the property s relation one derives the formula

s

s

:

= (27riu) --dT(s a

W i t h this

(--27ri)k+1 k r~kun (k.-1)!

'

'

Using that s "comlnutes" with shifts one finds a similar formula for + d)-k). Using those fornmlas for s one can show tile following: 1. Every b r C(u) with b(0) = b(oc) = 0 is the image of an element a E C(z) with ordo~ (a) _> 2. 2. s

= 0 if and only if there is a / C C(z) with f ( z + 1) - f ( z ) = a.

T h e last, statement has in fact two proofs. The first one observes that

f ( z + 1) - f ( z ) = a has a solution in C(z) if and only if a is the sum (over n and c~) of expressions of the tbnn:

jeZ

(z - f13+ J)'~ with ~

A calculation shows that s

j

flj = O. = 0 leads precisely to this formula for a.

10.1. C A L C U L A T I N G

CONNECTION

115

MATRICES

T h e second proof starts with the statement that /2(a) = 0 is equivalent to f (i.e. the element in the formal fundamental matrix) lies in C { z - ~ } . The equation f ( z + 1) - f ( z ) = a implies that f is then in C(z).

Example

10.7 The equation y(z + 1) = e2~i/my(z) + a with a E C(z).

Let us write r = e 2~rilm and e m = e(~'m). We suppose that a is chosen such that the equation has a formal solution f which is divergent. The choice a = z - 1 produces such an f. The equation reads in matrix form y(z+l)=

((,~ 0

a ) 1 y(z).

A symbolic fundamental matrix is

(0

eT?1

The two lifts

0

0

)

1

0

( 01 91 ) with g = e m .... l ( f ~ f t

f,.~ht).

1 The,nth

power o f g is an element

h E C(u). In general (for instance for a = z -z ) the equation X ~ - h is irreducible over C(u). This example shows that the finite extensions C ( u ~ ) with u ~ = u are needed for the description of the connection matrix. | P r o p o s i t i o n 10.8 Let y(z + 1) = A y(z) be a ~vgular singular equation with coefficients in C(z). Then the equation is equivalent (over C ( z ) ) to an equation where A has the f o r m Ze (: -

~)...(~

-

~)

(Ao +

A1 z - 1 q- . . . q- A e z - e ) ,

with ai E C* and Ao = 1. The connection matrix S of the equation is Ue

S =

[ I j ( 1 - e-2~i%a)

(B0 + B1 u - 1 q- . . . + B e U - e ) ,

with Bo invertible and Be = 1.

Proof." The first statement follows easily form the definition of regular singular. -1 F lelt. Let. F,.zyht and Flirt denote the two fundanaental matrices and so S = F,~i~ht T h e matrix Fright is an invertible holomorphic matrix for z with Re(z) > > 0.

116

CHAPTER 10. EXAMPLES OF EQ[L4TIONS AND GROUPS

Similarly for Fleft. P u t Gright := F(z - a t ) . . . F(z - ae)Fright. This is an invertible holomorphic matrix for z with Re(z) > > 0. Define atejr := I - I j { ( - 2 , ~ i e ~ ( z - a ' ) ) r ( 1 - (z - a j ) ) - l } f ~ e j , . T h e n a~ejt is holomorphic for z with Re(z) < < 0. Both Gright and Gleft satisfy the difference equation G.(z + 1) = (Aoz e + . . . A ~ _ l Z + Ae)G.(z). This implies t h a t G~eIt has no poles in C. T h e periodic matrix G~itghtGteft = I-[j(1 -e-2~i~,u)S has for z with Re(z) > > 0 no poles. Thus this expression is a matrix polynomial in the variable u. From S(0) = 1 and S(oc) is an invertible constant matrix one concludes that 1-Ij(1-e-2~a,u)S has the form ueBo+ue-lB,+...+Be. Moreover Be = S(0) = 1 and B0 = S(oc) is invertible. | We note that a somewhat similar statement is present in [9]. Birkhoff's fundamental problem is to show t h a t the map, which associates to (A0 = 1 , . . . , Ae) the ( B 0 , . . . , Be-l, Be = 1), is surjective. T h e fundamental problem asks in fact for the possibilities of the connection map. In Corollary 8.6 a complete answer is given for the case of regular difference equations over C(z). T h e answer, translated in the formulation of Birkhoff reads as follows:

Let B. := ( B o , B t , . . . , Be-l,Be) with Be = Bo = 1 be given. Then there is a ( A o , . . . , A e ) , with Ao = 1 and A1 = - ( a l + . . . + a e ) i d , which has image B.. One can extend this to the case of regular singular difference equations over C(z). Let T denote the neutral Tannakian category, for which the objects are the pairs (V, T) with V a finite dimensional vector space over C and T an autom o r p h i s m of C(u) | V such t h a t T(0) = 1 and T(c~) is an invertible constant map. In our terminology the answer to Birkhoff's question can be stated by the next result:

The morphism from the category of the regular singular difference modules over C(z) to the category T is an equivalence of tensor categories. In particular, every matrix T E Gl(n, C(u)) with T(O) = 1 and T(oo) is an invertible matrix, is the connection matrix of some regular singular matrix equation y(z + 1) = A y(z) with A E Gt(n, C ( z ) ) . T h e proof of Corollary 8.6 for the regular situation can be extended to the regular singular case. We will not give the details.

10.2

Classification of order one equations

Let k be a difference field. The order one equation r = ay is equivalent to r = by if and only if there is an f C k* with ab-x = r Let U(k) denote the subgroup of k* consisting of the elements ~@L)_,with f E k*. T h e group

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OF ORDER ONE EQUATIONS

117

of equivalence classes of order one equations over k is therefore equal to k * / U ( k ) . In this section we will consider for k the fields k ~ and ]%0 with r = z + 1. An easy calculation shows t h a t U(]%o) consists of the power series 1 + kz -1 + a2z-2+aaz-3+..., where k E Z and a s , h a , . . . E C. Every order one difference equation is therefore formally equivalent to a unique equation

y(z + 1) = z~c(1 + d z - 1 ) y ( z ) with n C Z, c C C*, d C C, such that 0 _< Re(d) < 1. We will show t h a t the classification of difference equations over koo is quite a different affair. This is unlike the (local) classification of order one differential equations, where the formal and the analytic theory coincide. Let the group H (with additive group law) be defined by the exactness of the following sequence: 6,

^

0 --+ H --+ k*~/U(koo) --~ koo/U(koo ) --+ 1. We introduce the following notations: ,, ocl is the unit circle, seen as the circle of directions at ec. 9 .A~ denotes the sheaf on S 1 of germs of 1-periodic meromorphic functions which are fiat at oc. One defines this as follows: Let (a,b) C S 1 be an interval, then an element ~ of .d~ is the germ of a meromorphic function f defined on a b o u n d e d sector at oc with arg(z) E (a, b) such t h a t f ( z + 1) = f ( z ) and f has 0 as asymptotic expansion on this sector. The following proposition describes the difference between analytic and formal classification of order one equations at ec. P r o p o s i t i o n 10.9 H is isomorphic to the complex vector space Ha (Sl,.Ape~).~ The complex vector space H is infinite dimensional. P r o o f : An element h of H is represented by some a E k ~ such t h a t f ( z + 1) -^, a f ( z ) has a solution f E k ~ . Write f = czkg with c E C*, k C Z and g(z-t- 1) __ 9 : 1 -t- g l z -1 + 92z -~ + .... T h e n h can also be represented by b :-- g-5~ry- l +b2z -2 +b3z -3 + . . . E k ~ . One writes b -- e x p ( B ) and g = exp(G), with B = B2z -2 + B a z - n .. . E z - 2 C { z -1} and G = G1 z - 1 + G2z -2 + . . . C z - l C [ [ z - 1 ] ] . The relation between B and G is the formula (r - 1)G = B. A small calculation shows that the m a p ( r 1 ) : z - l C [ [ z - 1 ] ] -+ z - 2 C [ [ z - 1 ] ] is bijective. It follows t h a t H can be identified with the cokernel of the m a p ((~-- 1) : z - l C { z

- 1 } ---+ z - 2 C { z - 1 } .

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118

Let B and G be as above. According to L e m m a 8.1, the formal power series G can be lifted to a unique meromorphic Gright satisfying:

1. Gright lives on a sector at oo with arg(z) C (-rr, 7r) and has G as asymptotic expansion on this sector. Let $1 C S t denote the interval (-rr, rr) of the circle of directions S 1 at oc.

2. G~ght(z + 1) - G~ijht(z) = B(z) holds in this sector. A similar asymptotic lift Gleft exists on a sector at oc, given by arg(z) r 0. Let $2 C S 1 denote the interval of the circle of directions at oo corresponding with this sector. W i t h an abuse of notation $2 is the interval (0, 2rr). The difference Gright -- Gleft is 1-periodic and has asymptotic expansion 0 on St A $2. Hence

Gright

--

Gtejt E

.A~

n

$2).

This element is seen as a 1-cocyele for the sheaf A~ on S 1 with respect to the covering {S~, $2} of S 1. Let a ( h ) denote the image of this 1-cocycle in the first eohomology group of the sheaf. Thus we have defined an additive m a p 1

1

0

o~ : H--+ H (S ,AVer). The sheaf A~

on S 1 is of a rather special nature:

1. On the interval arg(z) E (0, rr), the sheaf is constant with as stalk the complex vector space consisting of all holomorphic functions }-~--1 ane2~rinz converging on some upper half plane. Write again u = e 2rriz. T h e n this condition is equivalent t o E n % x anUn e t/C{'u}. 2. On the interval arg(z) C 0r,27r) (with abuse of notation), the sheaf is constant with as stalk the expressions ~-~2--~-1 ane2~rinz which converge on some lower halfplane. In other terms, A ~ is on (rr, 2rr) the constant sheaf with stalk u - X C { u - 1 } .

3. A~

= 0 if the interval S C S 1 contains one or both the direction arg(z) = 0 or the direction arg(z) = 7r.

On the intervals $1 and $2 and on their intersection $1 N $2, the sheaf A~ has trivial cohomology. The H I of the sheaf is therefore equal to the Cechcohomology with respect to the covering {$1, $2} of S 1. Thus

Hi(S1 , A poe r )

=

o Aver ($1 n

$2) ,

which is certainly an infinite dimensional vector space over C. Finally, we have to show that a : H -+ H I ( S 1, ~4per) 0 is bijective. If the 1-cocyele Gright -- Gle]t, corresponding to h E H is trivial, then Gright = Gleft

10.3. M O R E ON D I F F E R E N C E GALOIS GROUPS

119

and thus G E z - l C { z - 1 } . This implies h = 0. Hence c~ is injective. In order to prove the surjectivity of a we first consider the sheaf A ~ on S 1. This is the sheaf (of germs) of meromorphic functions on sectors at oo having asymptotic expansion 0. According to a result of Malgrange and Sibuya (see also Chapter 8), there is a natural isomorphism

C[[z-1]]/C{z -1 } --+ H I ( s l , . A ~

This isomorphism is induced by the map

/3: C[[z-1]] --+ H I ( s I , A ~ defined by/3(F) = (Fright -- Fl~jt), where Fright, Fie jr are meromorphic functions on the sectors S1 and $2 with asymptotic expansion F G C[[z-1]]. The difference (Fright -- Ft~ft) is a 1-cocycle for the sheaf A ~ with respect to the covering {S1, S~} of S 1. Let a 1-cocycle f E A~ N $2) be given. Then there is a F G C[[z-1]] with ~ ( F ) : Fright -- F l e f t is equivalent to f. One can change E,.i~ht and Fie jr by fiat functions such that actually Fright -- Fleft = f . For a fixed choice of F the Fright and Flirt are now uniquely determined by the condition Fright -- Fleft "~- f . The F itself is unique up to a change F + H with H E C{z-1}. The lifts of F + H are now Fright q- H and Fleit q- G and the satisfy again (Fright -k H) - (Flelt + H) = f. Let a 1-cocycle f for A~ be given. Since A~ is a subsheaf of A ~ there is a formal power series G, which we may suppose to lie in z - l C [ [ z - 1 ] ] , and there are lifts Gright, Glelt such that f = Gright -- Glair. Then G(z + 1) with lifts C-right( z -1- 1),Glelt(z q- 1) also satisfies Gright(Z q- 1) - Gtelt(z + 1) = f. Hence G(z + 1) - G(z) = h E z - 2 C { z - 1 } . It is clear from the construction that c~(h) is equal to the 1-cocycle f . | We remark Prgposition 10.9 is related to Corollary 8.6 and to the calculations in Example 10.5.

10.3

More on difference Galois groups

In Example 9.8 we have encountered an equation with difference Galois group Z / n Z x Z / m Z (both over koo and ]%o). The first factor comes from the decomposition of the Picard-Vessiot ring as a product of domains and the second factor comes from the finite field extension of koo (or ]r present in the Picard-Vessiot ring. We can make this more precise by a further analysis of the automorphism group Aut(R/]%o,r of the universal Picard-Vessiot ring _g = Ta[{e(g)}g~6,1]. We will use (for the moment) the notation G for this group of automorphisms. As noted in Chapter 6, G has the structure of an attine group scheme over C. For any difference module M (over C(z), k ~ or ]r the group G acts on

120

wo(M) = ker(r

C H A P T E R 10. E X A M P L E S OF E Q U A T I O N S A N D G R O U P S - 1, R @ M). T h e image of this action is the f o r m a l difference

Galois group, i.e. the difference Galois g r o u p of the m o d u l e 1 ~ | M . We int r o d u c e now two e l e m e n t s 7 and 6 of G . T h e i r irnages in the formal difference Galois g r o u p GM, Iormal of any M will be d e n o t e d by 7M a n d 5M. T h e choice for 3' a n d 5 is m a d e in such a way t h a t the images 7M and 5M have finite order in GM, f . . . . 1 and t h a t their images in aM, S . . . . . l/G~ ]ormat c o m m u t e and g e n e r a t e this group. T h e first element 3' is a l m o s t the same as the formal m o n o d r o m y . We define the action of 7 on "P by 7(z a) = e2~iaz a for all a E Q. T h e action of 7 on R is d e t e r m i n e d by 7(e(g)) = e(Tg ) and 7(l) = l. T h i s last choice makes 3' d i s t i n c t from the formal m o n o d r o m y . T h e Picard-Vessiot ring of any difference m o d u l e M over ]%0 is a s u b r i n g of R and is g e n e r a t e d over koo by a finite collection of elements: z a with a E Q; finitely m a n y elements e(g), with g E ~}; and p o s s i b l y I. F r o m this is it clear t h a t the i m a g e 7M in the formal difference Galois g r o u p of M has finite order. For the definition of the second element 5 E G , we need a h o m o m o r p h i s m e : C* --+ poo, where # ~ denotes the s u b g r o u p of C* consisting of the roots of unity. We require t h a t e is the identity on # ~ . Such a h o m o m o r p h i s m can be given by choosing, as before, a Q - l i n e a r s u b s p a c e L of C such t h a t C = Q O L. T h e m a p e is then defined by e(e 2~i(x+a)) = e 2~i~ for any A E Q a n d any a E L. T h e second a u t o m o r p h i s m ~ acts as the i d e n t i t y on P[l] and 5(e(g)) = h(g)e(g) with h : ~ --+ C* defined by h(zXc(1 + z-1)a~162 - q)) = e(c). It is easily seen t h a t the image of 5 in any formal difference Galois g r o u p has finite order. Let us i n t r o d u c e a s u b g r o u p G ~ of G as follows: G ~ consists of the a u t o m o r p h i s m s of R / ] ~ ( c o m m u t i n g with r which are the i d e n t i t y on P[{e(c)}]ce,~. T h i s g r o u p scheme G ~ has the p r o p e r t y t h a t its image in the f o r m a l difference G a l o i s group GM, format of any difference m o d u l e coincides with its c o n n e c t e d c o m p o n e n t of the i d e n t i t y G M, ~ ]ormat" T h e elements 7 and 5 do not c o m m u t e . T h e c o m m u t a t o r a : = 57(f-17 -1 is the i d e n t i t y on the s u b r i n g 7:'[{e(g)}ge~m,,~, l] a n d ae(z ~) = e2~iXe(zX). In p a r t i c u l a r , a E G ~. Hence the i m a g e of a in any f o r m a l difference Galois group GM, l . . . . . t lies in the c o n n e c t e d c o m p o n e n t G M, ~ Iormat" Moreover, if M is mild then the e(z ~) (for A r 0) are not p r e s e n t in the P i c a r d - V e s s i o t ring of M a n d as a consequence the images 7M a n d (~M c o m m u t e . F r o m the explicit form of R a n d the 7, d we come now to t h e following results on the f o r m a l difference Galois g r o u p GM, I . . . . l of a difference m o d u l e M over koo : 1 0 . 1 0 Let K denote the subgroup of GM, .format generated by the images ~/M and 5M of 7 and 5. Then:

Proposition

I. The map [~ -+ GM, ]ormal/G~ lor.~at is surjective and the group GM, ]ormat/G~ S . . . . l is commutative and has at most two generators.

10.3. M O R E O N D I F F E R E N C E

121

GALOIS GROUPS

2. I f M is mild, then the subgroup K C GM, .f . . . . . l is a finite c o m m u t a t i v e group with at most two generators.

Let M denote a difference module over ko~ or C(z). Let the Picard-Vessiot ring for M be denoted by P V and that of h'~ Q M by P V . There is an injective morphism f : P V ~ P V . This morphism is not unique, since it can be composed with a r automorphism of P V . However the image of f does not depend on the choice of f. For a fixed choice of f one finds an inclusion of the formal difference Galois group GM, formal --~ G, where G is the difference Galois group of M. This inclusion is unique up to conjugation by an element of G. In the sequel we will fix f and the inclusion GM, f . . . . . l ~ G. The elements ~M and (~M a r e now also considered as elements of G. Likewise, the subgroup K of GM, formal is considered as a subgroup of G. P r o p o s i t i o n 10.11

I. Let M be a difference module over C(z) with difference Galois group G. Then G / G ~ zs generated by the image of 5M.

2. Let M be a difference module over k ~ with difference Galois group G. Then the images o f dM and "/M in G I G ~ c o m m u t e and generate this group. 3. Suppose that M is a mild difference module over ko~. Then the subgroup K o f G generated by "/M and (~M is c o m m u t a t i v e and finite. Moreover K ~ G / G ~ is suryective.

P r o o f : 1. As above, the Picard-Vessiot ring of M over C(z) is denoted by P V . The ring P V is embedded in the Picard-Vessiot ring P~V of ~:~ | M over ]%~. Let G' C G denote the subgroup of G generated by G ~ and aM. The inclusion GM, formal C G implies G M, o formal C G ~ Let G'M, f or,~al denote the subgroup of GM. for,~al generated by G~ formal and 3M. Then G'M, f ...... l C G ~. Then GI

for the rings of invariants one has the following inclusion P V a' C P V M. S. . . . The last ring of invariants is a finite field extension o f / ~ . Since G I has finite index in G one has that P V c' is a finite extension of C(z). This finite extension lies in a finite extension of ] ~ and is therefore a field. The extension P V C' D C(z) is a finite extension of difference fields. This implies that P V c' is equal to C(z). By L e m m a 1.31, G = G I. 2. The Picard-Vessiot ring of M over ko~ is denoted by PV. The one of ]coo@ M by P~V. One fixes an inclusion P V C tffV. Let G' denote the subgroup of G generated by G ~ and "[M,(~M. As above one sees that GM, formal C G ~. The inclusion of the rings of invariants P V a' C P V GM' ~. . . . ' = / ~ implies that P V a' is a finite field extension of k ~ . This extension is moreover contained in k ~ . The conclusion is that P V c' = ko~ and thus G' = G. 3. We have already seen that 3'M and (~M commute if k ~ | M is mild. The rest of the statement follows from the previous parts of the proposition. |

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122

The following examples show that the map K --+ ( ; / G ~ can have a non trivial kernel. E x a m p l e 10.12 The order module M with equation y(z + 1) = (1 + z - 1 / n ) y ( z ) has the following difference Galois groups: (a)

over

(and over

c(z)).

(b) The cyclic group C~, generated by 7M, over the field k~o. The kernel of If -4 G / G ~ is generated by ~/M. P r o o f i (a) This mild equation is defined over C(z) and has G,~ as difference Galois group over C(z) according to example 10.2. Corollary 9.7 implies that its Galois group over k~ is also G,~. (b) Over ]%0 the equation is equivalent to the equation y(z + 1) = (1 + z - l ) 1/'~ A solution is z 1/" and the difference Gatois group is the cyclic group C,~, generated by 7M. | E x a m p l e 10.13 Let C, be a primitive nth root of unity. The equation y(z + 1) = ~ ( 1 + z-2)y(z) has difference Galois group G,~ over C(z) and over ko~. The difference Galois group over k'~ is cyclic and generated by 5M. Hence the kernel of K --+ G / G ~ is generated by 5M. Proof." As in the last example one shows that the difference Galois groups over C(z) and k~o are Gm. Over kc~ the equation is equivalent to y(z + 1) = ~ny(Z) and has therefore a cyclic difference Galois group generated by ~M.

10.4

Mild difference and differential e q u a t i o n s

We have seen that very mild differential equations and very mild difference equations over ]%~ form equivalent categories. This is no longer true if one replaces koo by kcc. A more precise result is the following. P r o p o s i t i o n 10.14 The formula r = e x p ( ~ ) induces a tensor functor .T from the category of mild differential modules over koo to the category of the mild difference modules over k ~ . The restriction of this functor to very mild differential equations is fully faithful, but not surjective on (equivalence classes of) objects. Proof: defines This is follows

Let M be an action equivalent that there

a mild differential module over koo. The formula q5 = e x p ( ~ ) of ~ on k ~ | M. We want to show that ~ is convergent. to showing that qS(M) C M. From the definition of mild it is a lattice M0 over C{z -1 } in the k~o-vector space M such

10.4. MILD DIFFERENCE AND DIFFERENTIAL EQUATIONS

123

that ~ M 0 C M0. One takes a basis el . . . . e~ of M0 over C { z - 1 } . The matrices B and C are the matrices of d and q5 with respect to this basis. The m a t r i x B has coefficients in C { z - z } and the matrix C has coefficients in C[[z-1]]. We have to show that the matrix C is convergent. One can expand ( d + B)~ as

a '~+, ~zz d ,~-1-~-" . .+*~+B(n). It is clear from the definition t h a t C d-7 We will use this formula to show that C is convergent.

:

~ ~B(n). .

For the B(n) there is a recurrence relation B(n) = B(n - 1 ) ' + B B ( n - 1) and B(1) = B. Write B(n) = ~ k > 0 B(n)k z-k" Let A >_ 2, c _> 2 be constants such t h a t ]lB(1)kll _< cA k. By induction one can show that

IIB(~)kll 5 (~ + 1)(k + 1)'~c'~Ak. Write C = ~ C k z -k then IlCkll _< E vergent.

~,llB(~)kll _< e2C(k+l)Ak.

Hence C is con-

The functor .T is defined as .TM is equal to M with the action o f ~ = exp(d). In order to prove t h a t 2- is fully faithful for very mild differential modules, it suffices to show that the m a p Hom(1, M) -+Hom(SCl, Y M ) is a bijection if M is a very mild differential module. The left hand side is equal to {m C M I ~ d =0} and the right hand side is {m E M[~m = m}. For a very mild difference module M over k~o the following formula holds in koo | M : d

=

log

( - 1 ) ~+1 (~

=

-

1)~.

n>0

This shows that the m a p is bijective. In the next example we will produce a regular difference module over koo which is not isomorphic to the .T-image of any very mild differential module over koo. | Example

10.15

The regular difference equation of order one, y(z + 1) = c(z)y(z) with c = 1 + , z -2 + , z -3 + ... E koo is the image under • of a regular differential equation y' = b(z)y for some b = , z -u + , z - a + . . . C ]%0. If b were an element of k~o then the differential equation y' = by is trivial, i.e. has a solution in k~o. Then y(z + 1) = c(z)y(z) has the same non trivial solution. However, we know t h a t a general regular order one equation does not have a solution in k ~ . We will produce a more explicit example. T h e relation between b and c is given in the proof of 10.14. In this special case this relation is b(z + 1) - b(z) = -~(~)! c(~)' The choice c(z) = e -z ~ produces the equation b(z + 1) - b(z) = - 2 z -3. The "connection matrix" of this equation

CHAPTER 10. EXAMPLES OF EQUATIONS AND GROUPS

124

is, according to 10.4 equal to d

.

d

-4rr2u .

~ ( - 2 z -3) = ~( ~ (z-2)) = ( 2 ~ , . ~ ) ( ~ ) t . ~- ~)- r 0. This shows that b is divergent.

10.5

Very mild difference modules and multisummability

The difference module M over koo is supposed to be very mild. With the notation of the proof of the last proposition, the action of ff on M is represented by a matrix C(z) and the action of the corresponding formal differential module is given by a matrix B(z) with coefficients in C[[z-1]]. The matrix B is the unique formal solution of the difference equation

B(~ + 1) - c(~)a(=)c(=) -~ = -(~c(:))c(=) -I This is again a very mild difference equation. The eigenvalues are the {gig~-1] where the {gi} are the eigenvalues of the equation y(z + 1) = C(z)-ly(z). In order to explain the behavior of the formal matrix B we have to recall some definitions and facts from the theory of multisummability.. We refer to [40] for more details. The general definition of multisummability is rather involved. The simple definition, given below, is in fact a theorem. Let k > 0 and let y = ~~1760 y,~z-~ be a formal power series. Let k > 0 and let d be a direction at oc. Then y is called k-summable in the direction d if there is a holomorphic function f defined on a (bounded) sector at oo with opening (d - a/2, d + a/2) and a > 88 and if there is a constant A > 0 such that for all N _> 1 and all z in the bounded sector the following inequalities hold N-1

If(z)- ~

y,~z-~l ~ ANF(1 + N)IzI-N.

rz=O

The condition posed on f is much stronger than saying that f has asymptotic expansion y. In fact, the holomorphic function f is unique and is called the k-sum of y in the direction d. We note that for k < 1/2 the f above is in fact a multivalued function defined on a sector with opening greater than 2rr. This difficulty can be removed by taking a suitable root of z. We refer to [40] for precise details in this case.

10.7. V E R Y M I L D D I F F E R E N T I A L M O D U L E S

125

For a sequence of positive n u m b e r s k = kl < . . . < kr and a direction d at, oc, the formal power series y is called k_.-nmltisumnmble in tile direct, ion d if y can be w r i t t e n as a s u m 9 = Yl + . . - + Y,. such t h a t each ,~i is k i - s u i n m a b l e in the direction d. Tile Yi are unique up to holomorphic functions at ,~. T h i s m e a n s t h a t we m a y change each //i into ~li + gi with 9i h o l o m o r p h i c at oc, (and with Y~gi = 0). Let fi be the k < s u m in the direction d then Y'~,fi is the multisum in the direction d. This m u l t i s u m is unique and lives as a function on a sector with o p e n i n g (d - cr/2, d + c~/2) with cr > ~ . A c c o r d i n g t.o [16] (see also T h e o r e m 11.1) the formal solution /3 is /~>multisummable in all b u t finitely many directions. T h e sequence k_ = k t < ( - 1 } of' the e q u a t i o n . . . < 1 are all the levels present in the eigenvalues {.qifqj

y ( : + 1) = C ( : ) - ~ ( : )

10.6

Very mild differential m o d u l e s

(1) Let (M, ~dz) be a very, mild differential m o d u l e with c o r r e s p o n d i n g very mild difference m o d u l e (M, ). T h e n (3I, ~ ) and (M, q5) have "the same asymptotic theory" for formal solutions. T h i s can be seen as follows. Let. r E k~. (q~ M be a solution of qs(v) = v. T h e n also ~-~, = 0. T h e asymptot.ic theory for differential equations asserts t h a t v is m u l t i s u l n m a b l e in a l m o s t all directions d at oo. Tile possible exceptions are the (finitely m a n y ) s i n g u l a r directions of (M, ~ ) . T h e positive slopes of the differential m o d u l e M are k__ = kl < . . . < k,, with k,. < 1 since the e q u a t i o n is very mild. T h e n v is k - m u l t i s u m m a b l e in almost all directions. Let. d be a direction for which v is k--multisummable. W r i t e v = vl + . . . + v,. as above; let fi be the ki-sum of vi in the direction d and put f = ~ fi. T h e n 0 ( f i ) is the /,'/-sum of d~(vi) in the direction d. Since v = ~ d;(vi) holds and since this d e c o m p o s i t i o n is unique up to convergent expressions at. ;x> we have (ui) = ui + .qi with (.qi E l]/I (i.e. the 9i are convergent) and ~gi = 0. T h e unicity of the ki-sunl implies t h a t ~)(f~) = ,f~ + g~. Hence Q(f) = f and so the mult.isuna of u in the direction d is a solution of the difference equation. (2) P r o p o s i t i o n 10.14 implies t h a t the wery mild differential m o d u l e (M,-g-d) de and the c o r r e s p o n d i n g very mild difference m o d u l e (M, ~ ) have the s a m e Galois group. (3) W'e will now discuss the p a p e r [12] of G.D. Birkhoff, which deals precisely with the difference e q u a t i o n g(z + 1) = (_' y(z) associated with a (very) mild differential e q u a t i o n y' = B y over ko~. Let. the invertible h o l o m o r p h i c m a t r i x Y ( z ) , defined on some sector at oc, satisf~y Y ( z ) ' = B Y ( z ) . T h e n Birkhoff s t a t e s

CHAPTER 10. EXAMPLES OF EQUATIONS AND GROUPS

126

that also Y(z + 1) = CY(z). This is correct and can be proved in the following way. One verifies that ( d ) ~ Y ( z ) = B(n)Y(z), ,,,here B(n) is the matrix introduced in the proof of Proposition 10.14. As a consequence the infinite expresd)'~Y(z) converges and has as sum CY(z). The sum is also equal to sion ~ ~(dz

Y(z + i). There is a fundamental matrix Y+(z) (this means u + 1) : CY+(z)), in the notation of [12], having the required asymptotic behavior on a sector ( - e - ~, ~ +e + -}).. Then Birkhoff claims that Y(z) = Y+ (z)D for some constant matrix D. In other words, Y+(z) also satisfies Y+(z)' = BY+(z). We will explain why this cannot be correct. Suppose that this is correct, then the same holds for the other fundamental matrix Y_ defined for a sector ( - e + ~, + e + ~-). The upper and lower connection matrices of the difference equation would then be constant matrices. Those constant matrices are the matrix 1 and the formal monodromy, since those matrices are their values at u = 0 and u = co. The conclusion from Corollary 9.6 is that the difference Galois group of Y ( z + 1) = C Y(z) is equal to the formal difference Galois group. The differential Galois group of 9' = By coincides, according to (2), with the difference Galois group of y(z + 1) = C y(z) and therefore with the formal difference Galois group of that equation. The last group is also the formal differential Galois group of y' = By. The final conclusion is that for any very mild differential equation y' = By at oc, the differential Galois group coincides with the formal differential Galois group. This is certainly not true. We will give an example. Consider the order two differential equation y(z)' =

z 1/2

0

0

__zl/2

)

y(Z).

After a transformation one can change this equation into a form defined over 1

--1/2z -1

9(z). The differential Galois group of

this equation is equal to the formal differential Galois group because we have chosen a canonical equation. This group is generated by the two dimensional exponential torus { ( * 0

0* ) } and a formal monodromy matrix ( 0 1

01 ) " A

small perturbation of the equation introduces Stokes matrices, which have the form either

0

1

or

The theory of,].-P. Ramis and J. Martinet 9

1

"

states that there is a small perturbation with a non-trivial Stokes matrix. Moreover, the differential Galois group is generated by the formal differential Galois group and the Stokes matrices. This implies that the differential Galois group of the perturbed equation is Gl(2, C).

Chapter 11

Wild difference equations 11.1

Introduction

The theme of this section is the problem of lifl,ing symbolic solutions of a wild difference equation to sectors or more general domains in C. The asymptotic behavior of the Gamma function is responsible for complicated analytic problems which do not occur in the case of a mild difference equation. The aim of "exact. asymptotics" is t.o find unique lifts with additional properties on certain sectors. Multisummation provides such unique lifts. F[ecent work of B.L.J. Braaksma and B.F. Faber [16] proves that, under certain hypotheses, symbolic (or formal) solutions are nmltisummable in many directions. We have already used their results for the analysis of the asymptotic theory of mild equations. As we will see, one cannot expect multisummability in the general ease (at least with the present definition of multisummability). In fact the rather restrictive hypotheses of [16] are necessary for multisummability. This brings us to asking for lifts of symbolic solutions which are not multisums and which are not unique. The important work of G.K. hnmink in [31, 32] will be the basis for our investigations. It is shown in [31, 32] that formal solutions can be lifted to quadrants. Our aim is to find large sectors at oo where formal solutions can be lifted. It is shown that on half planes formal solutions have asymptotic lifts. One uses a combination of two quadrants to prove the lifting property for a right half plane, more precisely on a sector of the form {: E CI - ~ < arg(z) <_ ~}. The case of a left. half plane is of course similar. Then same method proves the lifting property for the upper and lower half plane. The proofs use precise information about the asymptotic of the G a m m a function and other functions. Further a melhod, reminiscent of Cartan's lemma on analytic functions in several complex variables, is developed. We note that B.L.J. Bra.aksma and B.F. Faber are presently working at a direct analytic proof that half planes have the lifting property.

128

CHAPTER

11.

WILD DIFf'ERENCE EQUATIONS

T h e results above on the lifting p r o p e r t y for half planes lead to a definition of a "connection cocycle". T h e t h e o r e m of M a l g r a n g e and S i b u y a is the tool for the inverse p r o b l e m concerning this connection cocycle. This simplifies earlier results in this direction in [32].

11.2

M u l t i s u m m a b i l i t y of formal solutions

We will first give a formulation of the main result, of [16] which fits in our terminology. We want to define the singular directions of a difference e q u a t i o n 9(z + 1) = A y ( z ) with A an invertible m e r o m o r p h i c m a t r i x at z = oo. For convenience we will include + ~ in the set of singular directions. T h e other singular directions are related to the eigenvalues g E ~7 of the equation. Let g = zx e exp(r (14-Z-1) a~ be an eigenvalue. A direction o (always -Tr < 0 _< 7r) is singular for this g i r a function representing e(g) has locally m a x i m a l descent for z = re ~ and r --4 +,vc,. For each term s e p a r a t e l y we will calculate the s i n g u l a r directions. For c r 1. we can represent e(c) by the function e z(l~ If Icl -r 1 then one finds c o u n t a b l y m a n y singular directions having as limits the directions v~ , - ~ ~. If Ict = 1 then the only singular directions are 7 , ~ - T h e s i n g u l a r directions t h a t we have defined so far are called level i. T h e t e r m q is a finite sum ~0 0 a n d - ~ < r

~.

2. l f s = 1 and A1 = 0 . rr

3. I f As <_ 0 and either ~ < c3 < rc 07" -re < r <

~.

It is p r o b a b l e t h a t we have i n t r o d u c e d too m a n y singular directions a n d levels. T h i s is not essential for the a p p l i c a t i o n T h e o r e m 9.1 of tile t h e o r e m which is the

11.3. THE Q U A D R A N T THEOREM

129

basis of our analysis of the asymptotics of mild equations in Section 9. In the next example we will show that, multisummable of formal solutions is no longer valid in a more general situation.

E x a m p l e 11.2 Formal solutions which ere not multisummable The equation y ( z + l ) = z y ( z ) + l has a unique formal solution 9i E C((z -1)). According to Theorem 1 1.1, this formal solution is multisuinmable in the directions d = e'* with -g~ < 0 < v~ and with as nmltisum the function 9,.i~ht of Section 1 1..5. According to Sect.~oa 1 1.5, Yl is not nmltisummable in other directions. The equation g(z + 1) = g--ly(z) -t- 1 l~as a unique formal Y2- A similar analysis of this equation yields that 92 is nmltisummable precisely in the directions d = c ie) with g < 0 < We combine the two equations into an mhomogeneous matrix equation

92

(z+l)=

0

z -t

There is a unique f ~ 1 7 6 1 7 6

V:,

( g l' )9 2

1

'

This formalsolution is multi-

summable in a direction d if and only if both 91 and /)2 are multisummable in that direction. We conclude that the unique formal solution is in no direction multisummable!

11.3

The Quadrant T h e o r e m

A quadrant is a subset of tim complex plane of" the form

where v E C, k C Z and R > O. We will denote this quadrant by Q ( v , k , R ) . There are essentially four quadrants, they can be shifted over a complex number and a bounded part can be deleted. We note that changing the < signs in the definition of a quadrant into _< signs is not, essential for what, follows. A meromorphic function f on Q(v, k, R) i,~ said to have ~ > a a,z-'~/P E 79 as asymptotic expanszon if there is for every B a constant (7 stFch that kf(z) - ~B>~>_A a,.z-'~/r'[ <-- Clz[ -ts/r holds on Q(v,k,/~). The following theorem is proved by G.K. Birkhoff and W.J. Trjitzinsky in [14]. This proof contains

C H A P T E R 11. WILD D I F F E R E N C E E Q U A T I O N S

130

however a number of inaccuracies and it,s correctness has been questioned. In [31] and [32] a proof is presented by G.K. hnmink. The statement is the following. 11.3 Let the difference equation y ( z + l ) = A y(z), with A E Gl(n, koo) and koo is the algebraic closure of koo, be' given. Let A ~ be a canonical form for the difference equation. The formal matrix F E Gl(n,'P) 'with F ( z + I ) - I A F ( z ) = A ~ can be lifted to some invertible meivmorphie matrix F on Q(v, k, R) for a suitable R > O, i.e. F ( z + 1 ) - I A F ( z ) = A ~ and b' has asymptotic expansion [7, on Q(v, k, t~). Theorem

We sometimes prefer to work with the following equivalent form of the theorem: 11.4 Let {1 be a formal solution, i.e. with coefficients in 7), difference equation y(z + 1) = A y(z) where A E G l ( n , k ~ ) and k~ aigebraw closure of koo. Let v E C, k E Z also be giuen. There exists an and a meromorphic vector y on Q(v,k, R) such that y(z + 1) = A y(z) has ~1 as asgmptotic expansion in Q(v, k, I~),

Theorem

of the is the R > 0 and y

Proof." Let F be as in T h e o r e m 11.3. Using this, our equation y ( z + 1) = A y(z)

transforms under F to the equation v(z + 1) = A ~ v(z) with formal solution iJ. Since this equation is in canonical form, v is actually a constant vector. Transforming back with the inverse of F one finds the required Y on Q(v, k, R). T h e o r e m 11.3 is in fact equivalent to T h e o r e m 11.4 because F is a formal solution of the difference equation F(z + 1 ) - I A F ( z ) = A c. |

11.4

On the Gamma

function

We will need precise information on the asymptotic behavior of the G a m m a function. The classical result is r(z) ~

e-*e(z-ll')i~

1 -1 + ~:~z

+...},

for z with I arg(z)l < 2 - e and every positive e. The next lemma is concerned with the behavior in an upper strip. L e m m a 11.5 The Gamma fimction satisfies in an upper strip, i.e. a set of the for,,, {z E C I a < Re(z) < b, Ira(z) > c} for real numbers a,b,e and e > O, the inequalities

cll~l -~ _< Ir(~)ie ~t~'/~ ~ e~l~l ~, for certain positive constants cl,c2 and a positive integer n depending on the strip.

11.5. A N E X A M P L E

131

P r o o f : The formula F(z + 1) = zF(z) shows that, we can restrict ourselves to the strip with a = 0 and b = 1. The product formula reads

F(z)-* = ze'Y~ H {(l + Z ) e - ~ }. n> 1

We will use the notation f(z) "~ g(z) to denote t h a t there are positive constants el,e2 and there is a positive integer n such that. c~lzl -'~ < J(~) < e21zl ~ holds in the region where we the functions f and 9 are considered. W i t h this notation

F(z)-2~I~11+

[2~I-[(1+1

n>l

.~)2 sin(irclzl) )_ irclz[

n>l

2~1~1 ~&l~l The last expression is equal to ~<=1_~-~1=1

HenceF(z)~e

-~lzl/2.

R e m a r k s 11.6

(1) By conjugation z ~+ ~ one finds in a

~

1oW er strip" the same formula

ellzl -n ~ Ir(:)l err'z'/2 ~ c=lzl", for certain positive constants q , c~ and a positive integer n depending on the lower strip. (2) The behavior of the G a m m a function on the left half plane can be found from the behavior on the right half plane by means of the classical formula --Tr

r(z)

=

z sin(rcz)F(-z) "

In particular the G a m m a function has asymptotic expansion 0 on the two quad,'ants {z E CI Re(z) <_ O, Irn(z) > c} and {z E CI Re(z) < O, I'm(z) <_ - c } for any c > 0. We will not give the details of the proof.

11.5

An example

An analysis of the equation y(z + 1) = zy(z) + 1 will be our guide for the study of wild difference equations. It is clear that there is a unique formal solution ;0 C C ( ( z - 1 ) ) of the equation. In fact y C z - * C [ [ z - 1 ] ] . The asymptotic lifts on sectors at cc and on half planes is what we are studying. We start with a solution Y ~ h t of the equation. This solution can be found with the m e t h o d of L e m m a 8.I, namely

>,~ht = - ~ z-l(z + 1 ) - 1 n>0

(z + . ) - 1

CHAPTER 11. WILD DIFFERENCE EQUATIONS

132

T h e infinite s u m represents a m e r o m o r p h i c function on C with poles of order one at 0 , - 1 , - 2 , - 3 . . . . . W i t h the m e t h o d of L a m i n a 8.1 one can show that. ~l~'ight has 9 as a s y m p t o t i c e x p a n s i o n on the sect.ors (-~r + e, r: - e) for every positive e. W i t h more c o m p l i c a t e d e s t i m a t e s one <:an show t h a t ~lright is also an a s y m p t o t i c lift. of ~) on any u p p e r half plane (and any lower half plane) which does not contain the real axis. We want now t.o find solutions on the "left h a n d side". T h e q u a d r a n t t h e o r e m asserts t h a t there are a s y m p t o t i c lifts 92 and 93 on the q u a d r a n t s

#2 = {: c c l Re(z) < 0, ~,,~(~-) > - 1 } ~nd Q3 = {z E c [ ]~e(z) < 0, I r a ( z ) < 1}. T h e difference 92 - 9a is equal to h ( u ) F ( z ) on Q-, 21 Q3 where h is a merom o r p h i c function of u = e 2~i~, defined for e -2~ < lul < e 2~. F u r t h e r m o r e 92 93 has a s y m p t o t i c e x p a n s i o n 0 on Q2 C~Qa- Ill p a r t i c u l a r for z E Q2 M Qa and Re(z) < < 0, the function 9-~ - Y3 has no poles. T h a t implies t h a t h has no poles. Also h(1) = 0 since F(z) has poles in 0 , - 1 , - 2 , - 3 , . . . . Hence h = ~ _ _ _ ~ , : h,~ u ~ and this expression c o n v e r g e s oil e -2~r < [u.[ < e ~'r. Put. h + := a + ~ z > 0 h,~ttT~ and h - := b + ~,~.<~ h,~u". T h e c o n s t a n t s a, b are chosen such t h a t h+(1) = h - ( 1 ) = 0. Since h ( l ) = 0 we have h = h + + h - . We c l a i m t h a t h + (lz)F(z) has a s y m p t o t i c e x p a n s i o n 0 on O2. W r i t e h+(,L) = (u - 1)h('u). T h e n k(u) is b o u n d e d for 0 < lul _< R and any R < e 2~. T h e function (u - 1)P(a) has no poles. We have to estinaate (u - 1)r(z) on 02. On the b a n d {z E CI - 1 _< Ira(z) <_ 1, Re(z) <_0} the function ( I t - 1)F(z) has a s y m p totic e x p a n s i o n 0 as can be seen from the f o r m u l a (,u,- 1)r(z) = z-~r(e=sin(Tra)P(-z) "1) On tile q u a d r a n t {z E C] Re(z) <_O, I'm(a} > 1} the function ( u - l ) is b o u n d e d and the G a m m a function has a s y m p t o t i c expansion 0. (See R e m a r k s 11.4, p a r t ( 2 ) ) . T h e s a m e a r g u m e n t s show t h a t h - ( t t ) F ( : ) has a s y m p t o t i c e x p a n s i o n 0 on ,q2 h+(u)F(z) gild t]3 -t- h-(~t)F(z) are a s y m p t o t i c lifts of 9 on Q2 and Q3. By c o n s t r u c t i o n y~ - h+(-u)F(z) = Y3 + h - ( u ) F ( z ) on Q~ 21Q3- T h u s we have found an a s y m p t o t i c lift 9z~rt of 9 on the left. half plane Q2 u Q3. T h e function 9left is m e r o m o r p h i c on C since it is a s o l u t i o n of our e q u a t i o n 9(z + 1) = zg(z) + 1. S u p p o s e t h a t f is also an a s y m p t o t i c lift of 9 on Q2 u Q3. T h e n tile difference Yle.[t - - f has tile form k(u)F(z) where k is a h o l o m o r p h i c function of u, defined for u E C ' . Moreover the e x p a n s i o n ofk(u)F(z) in Q2uQ3 is 0. On an u p p e r s t r i p we use the e s t i m a t e of L a m i n a 11.5 for the G a m m a funct.ion and we conclude t h a t k has no pole at u = 0. S i m i l a r l y k has no pole at. u = oc,. T h u s k is a c o n s t a n t . T h i s c o n s t a n t is 0 because the G a m m a function has poles ill 0 , - - 1 , - - 2 . . . . . T h i s shows the unicity of the a s y m p t o t i c lift ~]left Q:3. T h e n

11.6.

SOLUTIONS ON A RIGHT HALF PLANE

133

on Q2 u Q3- One can find Yle:t explicitly. The function Yle]t Yright has the form h(u)F(z) with h a holomorphie function of u C C*. The function h(u)F(z) is asymptotically 0 in an upper strip. The behavior of F in this strip implies t h a t h has no pole at u = 0. Similarly, h has no pole at u = oc. T h e n h is a constant and Yl~:t = Y r i g h t -t- hF. The flmction Yleft has no pole at z = 0. This determines the constant h. --

Let f be an asymptotic lift of 9 ill a sector at oc which has a non e m p t y intersection with the sector ( - ~ , ~), then on the intersection S of the two sectors the function f - Yright has a s y m p t o t i c expansion 0. Write f - Yright k(u)F(z) where k is meromorphic function of u, which is holomorphic for z with tzl > > 0. If S contains the direction arg(z) = 0 then it is rather clear t h a t k = 0. If S lies in (0, ~) then k is holomorphic for u with 0 < lul < ~ with some positive 6 and has a Laurent expansion k = ~n~ k , u n. For t G S such that also t + 1 E S =

and any m E Z one can form the integral at f t + l k(u)u_ m dz. This integral is equal to k,~. T h e integrand can be estimated on the interval [t, t + 1] by for some constant c~> 0. By shifting t E S to oc one obtains km = 0. Hence k = 0. An analogous reasoning shows that k = 0 if the sector S lies in ( - ~ , 0). We conclude that the assumption on f above implies t h a t f = Yright. For an open sector S we say t h a t f is an asymptotic lift of ~) if f satisfies the equation and has ~) as a s y m p t o t i c expansion on every closed subsector of S. T h e 71- 37r two sectors (-Tr,~r) and (7, Y ) (abusing the notation) are then the two maximal open sectors on which 9 has an asymptotic lift. The lifts are Yri9ht and Yt~yt. Suppose t h a t ~) is multisummable in the direction d = e ir with ~ _< r < ~-. The multisum Yd in t h a t direction has the correct asymptotic expansion on an open sector S containing [r - 2, r + 5]~ because the only level present is 1. This contradicts the statement about the maximal sectors where ~) has an a s y m p t o t i c lift. We conclude that ~) is not nmltisulnmable in the direction d. T h e m e t h o d of this example will be used in the next sections for the construction of asymptotics lifts of formal solutions in the general situation.

11.6

S o l u t i o n s o n a right h a l f p l a n e

The following result is quite close to T h e o r e m 18.13 of [30].

T h e o r e m 11.7 Let {I be a formal solution, i.e. with coefficients in 7~, of the difference equation y(z + 1) = A y(z) where A E G l ( n , k ~ ) and k ~ ,s the algebraic closure of kcr There exists a meromorphic vector y on a right domain V such that:

C H A P T E R 11.

134

WILD DIFFERENCE EQUATIONS

1. y ( z + l ) = A y ( z ) 2. y is holomorphic for z E C with Re(z) > > O. 3. There is a real number b such that for every ~ > 0 the restriction of y to 71"

{z E v[ Re(z) > b and - ~ + ~ _< a~g(~)

71"

< ~)}

has ~1 as asymptotic expansion. Proof." As in the proof of T h e o r e m 11.4, it suffices to prove this theorem for a formal solution F of the equation f~(z + 1 ) - I A F ( z ) = A c, where A c is the canonical form. For the proof of the latter we prefer to work with modules. Let M be the module corresponding to A and M c the module corresponding to A ~. Then F corresponds to an isomorphism (r : P | M --+ 7) | M ~. We take two quadrants Q1 and Q4, with k = 0 and k = - 1 , which cover a right half plane and have as intersection a band B : = {z E CI llm(z)l < cl, Izl > c2} for certain positive numbers cl, c2. We m a y suppose t h a t (r has asymptotic lifts or1 and cr4 on the two quadrants. The m a p ~ 4 a l I is an a u t o m o r p h i s m of M ~ above this band B and is asymptotically the identity. In the next l e m m a we will show t h a t there are a u t o m o r p h i s m s rl and r4 of M c such that: 1.

7"1

is defined above Q1 and is asymptotic to the identity on Q1.

2. r4 is defined above Q4 and is asymptotic to the identity on {z E Q4[ [zl >_ R, are(z ) E ( - ~ + e, 0)} got' a fixed R and all e > 0. 3. ~4~

I = r[lrl.

Then we change the lifts ~r1 and o4 into r1~, and r4c%. The new lifts coincide on the band and glue therefore to a lift O'right of o" with the a s y m p t o t i c behavior required in the theorem. Going back to matrices, we have found a meromorphic F on a right half plane with F ( z + 1 ) - Z A F ( z ) = A c and with a s y m p t o t i c expansion F on the regions described in the theorem. The equation satisfied by F shows t h a t F is in fact defined on a right domain. The a s y m p t o t i c behavior of F implies that F is holomorphic for z with Re(z) > > 0. The a s y m p t o t i c behavior of F -1 shows that F -1 has no poles for z with Re(z) > > 0. Hence F is an invertible holomorphic matrix for z with Re(z) > > 0. This shows that our solution y of the original problem is holomorphic for Re(z) > > 0. | We suppose now t h a t the difference module M over the algebraic closure koo of koo is in canonical form. This means that M is a direct sum of modules E(9) | Mg where g runs in a finite subset of G, where E(9) = kooeg with r = ge 9 and where Mg is a unipotent module, i.e. the operation of(P has on a special basis the matrix 1 + N l o g ( 1 + z -1) with N a nilpotent matrix.

11.6. S O L U T I O N S ON A R I G H T H A L F P L A N E

135

11.8 Suppose that M is a difference module in canonical form. Let an automorphism r of M above the band B be given which is asymptotically the identity. Then there are automorphisms rl and r4 of M above the quadrants Q1 and Q4 such that:

Lemma

1. 7"1 is asymptotically the identity above Q1. 2. r 4 is asymptotically the identity for z C Q4, Izl > R, with a fixed R, and

arg(z) E ( - ~ + e,O) for every e > O.

3. r = r41rl . P r o o f : In order to simplify the notations we will suppose that every Mg is one dimensional. In other words, we suppose that M has a basis e l , . . . , e ~ over the algebraic closure of koo such that r = 9jej where g l , . . . , g, are distinct elements of ~. For g E ~ of the form g = z'e2~:i(a~162 - q)(1 + z - l ) b, with 0 _< a0 < 1 and al E R, we define e(g). = F(z)~e>~i(a~ b. For 9 = 9j all the items in the formula for e(g). will be indexed by j. The a u t o m o r p h i s m r has the form r(ek) = ~ k hJ.k(u)e(gj)21e(gk)*eJ, with each hj,k(u) a holomorphie function of u = e 2~z in the domain {u E CI e -2~r < lul < e >~< }. The condition that r is asymptotically the identity translates in to hj,j - 1 and hj,k(u)e(gj)21e(gk)., with j -r k, have asymptotic expansion 0 on the band. It follows at once t h a t hj,j = 1. For j :fi k, one finds inequalities [hj,k(u)[ <_ cNle(gj).e(9~:),lz N ] for every positive integer N and with a positive constant CN depending on N. We choose now an ordering of the basis e l , . . . , e~ such that for j < k one of the following statements is correct 1. Aj < Ak. 2. ~3 = kk and al(j) > al(k). 3. ~J = ~k, a l ( j ) = al(k) and if there is a # with Re(a,) -J: 0 in the expression ~ auz" := qj - qk then the highest # with Re(a,) :/: 0 satisfies

_Re(a,) < O. It is clear that such an ordering exists, The G a m m a function has on the band B the asymptotic behavior: F(z) ~ e-z e(z-1/2)l~

~ z -1 q- ...}.

Using this one finds t h a t hj,k = 0 for j < k. Hence the matrix of r is an upper triangular matrix with l ' s on the diagonal. Our task is to write r = 7"417-1 with a u t o m o r p h i s m s r4 and rl which are asymptotically the identity above {z C Q41arg(z) c ( - 2 + e,0)} (every e > 0) and Qa. We do this by multiplying the matrix of r on the right and on the left by a sequence of upper triangular matrices, with l's on the diagonal, coming from a u t o m o r p h i s m s which

C H A P T E R 11. WILD D I F F E R E N C E E Q U A T I O N S

136

are asymptotically the identity above Q4 and (21. In the first step one wants to kill the entries of the m a t r i x of r which are on the line above the diagonal. Each of the following steps should remove the entries on a line parallel to the diagonal. One sees that it suffices that to solve the following "additive problem ""

Let j > k and let h(u)e(gj)jle(g~), have asgmptotw ezpansion 0 on the band 13. Then h can be written as a sum. of h + and h - such that: (1) h+(u)e(gj)jle(gk), has asymptotic expansion 0 on Q1. (2) h - ( u ) e ( g j ) j l e ( g k ) , has asymptotw expansion 0 on {z E Q4I - ~ 4- ( < arg(z) _< 0} for all( > O. Put h = ~,,~176 c~u ~. T h e first case to consider is Aj > Ak. Then h + := ~ > A c, u~ and h - := }-~,~
{z E C I a < Re(z) < b, Ira(z) > c} for real numbers a,b,c with c > 0. Using the estimate of L e m m a 11.5 for the G a m m a function, one finds that on this strip the function e(gj)sle(gk), behaves like U()'k-)'J)/4+a~176 -- qj). T h e choice A = - { ( ~ k - .Xj)/4 + ao(k) a0(j)} guarantees t h a t h + satisfies (1). Tile a s y m p t o t i c formula for the G a m m a function for z with arg(z) E ( - ~ , ~ ) i m p l i e s that h - satisfies (2). T h e next case to consider is Ak = ~j and a l ( k ) > a l ( j ) . condition that U ne2r:i(a~176

Let n E Z. T h e

) )~exp(qk - qj)

is a s y m p t o t i c a l l y 0 for 0 _< arg(z) _< ~ is equivalent to

(n + ao(k) - ao(j)) sin(C) + (al (k) - a l ( j ) ) cos(C) > 0 for 0_< r ~. This is the case i f n + a 0 ( k ) - a 0 ( j ) > 0. T h e condition t h a t the same expression has a s y m p t o t i c expansion 0 for - ~ + c < arg(z) <_ 0 (every positive ~) is true if

(n + ao(k) - ao (j))sin(C) + (a 1 (k)

-

-

al (j))cos(C) > 0

for - ~ < r < 0. This is the case if n + ao(k) - ao(j) < O. From this it follows that the choice h + = ~,~>~o~j)-ao(k)c~ u'~ and h - = ~<_~o(d)-~0(k)c~u'~ has the correct a s y m p t o t i c properties. T h e last case to consider is Ak = Aj, al(k) = ai(j) and "if }-~ aaz" := qk - q j has a t e r m with R e ( a . ) # 0 then the largest # with Re(a~) # 0 satisfies Re(a~,) <

11.7. SOLUTIONS ON A N UPPER HALF P L A N E

137

0". Let n E Z. The expression u~e(gj)21e(gk), has asymptotic expansion 0 on 0,1 i f n + a o ( k ) - a o ( j ) > 0. The same expression u~e(gj)21e(gk), has asymptotic expansion 0 for - ~ < arg(z) _< 0 i f n + a o ( k ) - a o ( j ) < 0 as one can easily verify. 7r 7r Suppose that n + ao(k) - ao(j) = 0 then either for 0 _< arg(z) _< 7 or for - g < arg(z) _< 0 the expression u~e(gj),le(gk), has asymptotic expansion 0. As in the previous case the choice h + = ~ > ~ 0 ( j ) - ~ 0 ( k ) e~u'~ or h + = Y'~n>_~o(j)-ao(k) c'~u'~ and h - = h - h + has the correct asymptotic properties. I R e m a r k s 11.9 (1) The m e t h o d s of the proof of T h e o r e m 11.7 allows the following variation on T h e o r e m 11.7:

There is a solution y of the equation, defined on a right domain, holomorphic for Re(z) > > 0 and with 9 as asymptotic expansion on sets 7r

7r

for a fixed R and all e > O. (2) There are two obvious analogues of T h e o r e m 11.7 for left half spaces. (3) One might think t h a t a somewhat larger sector than a half plane has already the "lifting property". Section 11.5 shows that this is not the case. Indeed, the solution Yleft does not have ~) as asymptotic expansion on, say, the sector 71 371 (y - e, -5-) for e > O. (4) Also an extension of T h e o r e m 11.7 to the closed sector [ - 2 , 2] is in general false. This can easily be seen from the "additive problem" stated in L e m m a 11.8.

11.7

Solutions

on an upper half plane

The next t h e o r e m gives a positive answer to the old question: "Does a formal solution of a difference equation lift to sectors around the directions _y.4~ 9-. We will show that a formal solution 9 can be lifted to a solution y on an upper plane H = {z E C I Ira(z) > c}. More precisely, y has asymptotic expansion on {z E HI 0 <_ arg(z) < ~r - e} for every positive e. The obvious variant of this: there is a solution y such that y has y as asymptotic expansion on {z E HI r _< arg(z) < ~r} for every positive e is equally true. T h e stronger s t a t e m e n t that y has the correct asymptotic expansion on the whole of H is probably false. Section 11.5 shows at least that one cannot expect an a s y m p t o t i c lift on an open sector which is strictly greater than (0, rr).

C H A P T E R 11. WILD DIFFERENCE EQUATIONS

138

1 1 . 1 0 Let ~1 be a formal solution, i.e. with coefficients in 7), of the difference equation y(z + 1) = A y(z) where A E Gl(n, koo) and k~ is the algebraic closure of k~. There exists an upper half plane H := {z E C[ Ira(z) > c} for some c > 0 and a meromorphic vector y, defined on a right domain containing H, such that:

Theorem

1. y(z + l) = A y(z). 2. y is holomorphic for Re(z) > > 0. 3. For every ~ > 0 the vector y has ~] as asymptotic expansion on {zEH 10_<arg(z)_<~r-e}. P r o o f : The proof has the same structure as the proof of T h e o r e m 11.7. We have in fact to prove the analogue of L e m m a 11.8. Let a canonical difference module M over the algebraic closure of koo and an a u t o m o r p h i s m r of M, defined and asymptotically the identity on an upper strip {z E C] a < Re(z) < b, Im(z) > c}, be given. We have to show that r = rlr~, where the a u t o m o r p h i s m r~ lives on a left domain and must for every positive e be asymptotically to the identity for {z E HI Re(z) < b} and 0 <_ arg(z) < ~r - e. Furthermore the a u t o m o r p h i s m r~ must also live above a right domain and must be asymptotically the identity on the set {z E HI a < Re(z)}. For notational convenience we suppose that the canonical module M has a basis el . . . . , e~ over the algebraic closure of k~ such t h a t r = gjej, where 91, gs are distinct elements of G. For an element 9

9

9

,

g=zXe~(a~162

-- q ) ( l + z - ~ )

b ~ G,

with a0, al E R and la01 < 1, we write e(g), z ['(z)Xe27ri(ao+ial)Zexp(q)zb. As in Section 11.5, one can show that if ( ~ = - c ~ hnu'~)e(g) * is asymptotically 0 on the strip then each term h,~u~e(g), has also this property. Our first concern is therefore to find out when u"e(g), is asymptotically 0 in the strip. Using the estimate of L e m m a 11.5 for the G a m m a function in the strip, one finds t h a t the integers n are given by: (a) g + a 0 + n > 0 a n d (b) g + a0 + n = 0 if q has the special property (*), P r o p e r t y (*) is the following. Write q = ~ a,zt`. If there is # with Re(at, i" ) • 0 then the largest tt with Re(at`i") :/: 0 satisfies Re(at`i") < O. In the next calculation we want to know whether an expression u'~e(g), which is asymptotically 0 on the strip is also asymptotically 0 on {z E C] Irn(z) > O, Re(z) < b, 0 < arg(z) < z r - e} for every positive e. T h e last property will be called left-fiat. The results are the following:

11.7. SOLUTIONS ON AN UPPER HALF PLANE 1. For A < 0 the expression

139

u'~e(g), is not left-fiat.

2. For A > 0 the expression u'~e(g), is left-flat if gA + a 0 + n > 0. expression is also left-flat if g + a0 + n = 0 and q satisfies (*).

The

3. For A = 0 and a l < 0 the expression is left-flat if a0 + n > 0 a n d also for a0 + n = 0 (in t h a t case a0 = n = 0) if q satisfies (*). 4. For X = 0 a n d a l > 0 the expression is

not left-fiat.

. For A = 0 a n d a l = 0 the expression is left-flat if a0 + n > 0. If a0 + n = 0, a n d so a0 = n = 0, the expression if left-flat if q has the p r o p e r t y Re(q(z)) < 0 for z with Ira(z) > 0 and : <_ arg(z) < rr. T h i s p r o p e r t y of q will be d e n o t e d by (*1). We will call a function right-flat if the function has a s y m p t o t i c e x p a n s i o n 0 on the set {z E CI Ira(z) > 0, Re(z) > b}. Let the expression u'~e(g), have a s y m p t o t i c e x p a n s i o n 0 on the strip. T h e question w h e t h e r une(g), is right-flat has the following answer: 1. For A > 0 the expression

u'~e(g), is not right-flat.

2. For A < 0 the expression u'~e(g), is right-flat if g + a0 + n > 0. expression is also right-flat if ~ + a0 + n = 0 and q satisfies (*).

The

3. For A = 0 and a l > 0 the expression is right-flat if a0 + n > 0 a n d also for a0 + n = 0 (in t h a t case a0 = n = 0) if q satisfies (*). 4. For A = 0 and a l < 0 the expression is

not right-flat.

. For A = 0 a n d a l = 0 the expression is right-fiat i f a 0 + n > 0. If a 0 + n , and so a0 = n = 0, the expression if right-flat ifq has the p r o p e r t y Re(q(z)) < 0 for z w i t h I r a ( z ) > 0 a n d 0 < arg(z) < 9" T h i s p r o p e r t y o f q will be d e n o t e d by (*r). One can easily a n a l y s e the p r o p e r t i e s (*),(*l) and (*r). I f q satisfies (*) then q satisfies (*l) or (*r) (or b o t h ) . This shows t h a t ifu~e(g), a s y m p t o t i c a l l y 0 on the s t r i p then u'~e(g), is left-flat or right-flat (or b o t h ) . For the g l , . . - , gs E ~ above we choose the e(gj), as above b u t now with the c o n d i t i o n 0 <_ ao(j) < 1. T h e ordering of the basis e l , . . . , es is chosen such t h a t j <_ k implies t h a t e(gj),le(gk), has the p r o p e r t y :

If une(gj),le(gk), is asymptotically 0 on the strip then u'~e(gj),le(gk), is left-fiat.

C H A P T E R 1 I. WILD D I F F E R E N C E E Q U A T I O N S

140

F r o m the results above one deduces t h a t such an o r d e r i n g exists. T h e given a u t o m o r p h i s m r of M above the s t r i p c o m m u t e s with r and has therefore the matrix T( k) =

J where the hj,k are h o l o m o r p h i c functions of u for 0 < lu] < ~ for a c e r t a i n 6 > 0. For j r k one has t h a t hj,k(u)e(gj)-21e(gk), is a s y m p t o t i c a l l y 0 on the strip. Also hj,j - - 1 is a s y m p t o t i c a l l y 0 on the strip. It follows t h a t the hj,k have at m o s t poles at, u = 0. In o t h e r words the functions hj,k belong to the field C ( { u } ) . T h e m a t r i x of v is a s y m p t o t i c a l l y the i d e n t i t y on the strip. F r o m this it follows t h a t for any r with 1 < r < s the d e t e r m i n a n t of the m a t r i x (hj,k(u)e(gj)-~le(gk).)lfj,k<_r is non zer--o. T h e m a t r i x (hj,k) inherits this p r o p e r t y , i.e. for every r with 1 < r < s the d e t e r m i n a n t det(hj,k)l<_j,k
1. (aj,k)(bj,k)= (hj,k). 2. aj, k = 0 if j > k. 3. bj,k = O if j < k. 4. bj,j = 1 for all j . Define rl a n d r~ by the m a t r i c e s rl(ek) = Z aJ,k(u)e(gj)*le(gk)*eJ J

and

J C l e a r l y ~r~ = r . T h e orders of the functions hj,k a t the p o i n t u = 0 are known. F r o m this one can c a l c u l a t e the orders of the functions aj,k a n d bj,k at u = 0 a n d verify t h a t rl and r~ have the required a s y m p t o t i c b e h a v i o r . |

Theorem 11.10 has two analogues for a lower half plane.

Remark

11.11

11.8

Analytic equivalence classes of difference equations

In this s u b s e c t i o n we s t u d y the set of difference e q u a t i o n s which are f o r m a l l y equivalent to a fixed difference e q u a t i o n y ( z + 1) = S y(z). T h e invertible m a t r i x

11.8. A N A L Y T I C E Q U I V A L E N C E CLASSES

141

is supposed to have coefficients in the algebraic closure of/ace. ~,Ve consider the pairs (A, F) with the properties: 9 A is an invertible matrix with coefficients in the algebraic closure of koo. 9 F is an invertible matrix with coefficients in the field of Puiseux series 7).

9 F(~" + 1 ) - ~ . 4 F ( ~ )

= S.

For such a pair, the equation 9(z + 1) = A y(z) is formally equivalent to the equation y(z + 1) = S y(z), and F is a choice for the formal equivalence. On this set of pairs we introduce also an equivalence relation. The pairs (Ai, Fi), i = 1,2 are called equivalent if there is an invertible meromorphic matrix C, defined over the algebraic closure of k~, such that C(z + 1)-IAIC(Z) = A2 a n d F1 = CF2. The set of equivalence classes will be denoted by Eq(S). The idea for the study of Eq(S) is to lift the formal F on sectors at cxD and to compare the various lifts on the intersection of the sectors. This comparison leads to a cocycle for a certain sheaf of groups Aut(S) ~ on the circle S 1 of the directions at oc. This sheaf is defined as follows: Let (a,b) be an open interval of ~,1 then Aut(S)~ consists of the invertible meromorphic matrices T, defined on a sector at oc corresponding to (a, b), such that T has asymptotic expansion 1 for every closed subsector and such that T(z + 1)-~ST(z) = S. For the convenience of the reader we recall the definitions for the first cohomology set of a sheaf of nor~ abelian groups. Let 6'1,..., S,~ be a covering of the circle S 1 by open intervals. A cocycle with respect to this covering is a family of elements {{i,j}, such that 9 Each {i,j is a section of the sheaf Aut(S) ~ above Si N Sj. j,a

"

9 On Si N Sj n Sk the equality ~i,j~#,a.~i = 1 holds. The trivial cocgcle is the cocycle with all ~i,j = 1. Two cocycles {~i,j} and {q<j} are called equivalent if there are sections Ui of our sheaf above Si such that qi,j = u(-z~i,jUj for all i,j. The set of all equivalence classes of cocycles with respect to the given covering of S 1 is the first Cech cohomology set of the sheaf with respect to this covering. The direct limit, over all coverings of the circle, of those C'.ech eohomology sets, is the first C_2echcohomology set of the sheaf on S 1. This set will be denoted by H i ( S ) . We note that the map from the cohomology set with respect to a fixed covering to H 1(S) is injective. It suffices to consider coverings $ 1 , . . . , S~ by open intervals of the circle, such

C H A P T E R 11. WILD DIFFERENCE EQUATIONS

142

that there are no triple intersections. One may then suppose that Si has only a non trivial intersection with S i - i and Si+I, where we have introduced the convenient "cyclic" notation Si = Si+k~ for any k E Z. A coeycle can now be represented by (~i,2, ~2,3, . . . , ~ - 1,,~, ~,~,1). There are no longer conditions on the elements ~i,i+l. For the special situation that concerns us, it suffices to consider the covering {S1,$2,$3,$4} given by the open intervals (-~-2 ~ 2~-) (0, rr), (~, T 3~) and (-rr, 0). For each Si and each open covering of Si, one can show that any cocycle is equivalent with a trivial one. This is a consequence of the lifting property for the sectors Si. It follows that every element of H i (S) can be represented by a cocycle with respect to the covering {Si, $2, Sa, $4}. In other words, we find the following description of H i (S): H 1(S) is the set of equivalence classes of 4-tuples TLi+i is a section of the sheaf Aut(S) ~ Two 4-tuples {Ai,i+l } are {Bi,i+i } are equivalent if there are sections Ui of the sheaf above Si such that B i , i + 1 = U[-1Ai,i+lUi+l for all i.

(T1,2, T2,s, T3,4, T4,1), where

Theorem

11.12 There is a natural bijection Eq(S)

--+

Hi(S).

P r o o f i Let a pair (A, fi) be given. We know that there are asymptotic lifts F~ of fi on the sectors Si for 1 = 1 , . . . , 4. We associate to the pair (A,/~) the cocycle (F~IF2, F~iF3, F a l F 4 . F 4 i F i ) . The class of this cocycle does not depend on the choice of the Fi. Equivalent pairs produce equivalent cocycles. Hence there is a well defined map Eq(S) -0 H i(S). The injectivity of this map is an easy exercise. The surjectivity however is not at all clear. We will use the theorem of Malgrange and Sibuya to prove this. Let a cocycle (Ti.2,T2,3,T3,4, T4,1) be given. This can also be seen as a cocycle for the sheaf H introduced in Section 8.4. Hence there is a F E G L ( n , C ( ( z - i ) ) ) and there are lifts Fi o f / ? such that FZIF~+I = T~.~+i for all i. The matrices Ai = F~(z + 1)SF~(z) -1 are invertible meromorphic on the sectors Si. On the intersections Si A Si+I the equality Ai = Ai+l holds (by construction). The Ai glue to an invertible matrix with coefficients in the algebraic closure of koo. The pair (A, F) has obviously the same image in H i (S) as the cocycle (T1,2, T2,3, T3,4, T4,1). | C o r o l l a r y 11.13 Suppose that the coefficients orS are in C(z). Let the cocycle (T1,2, T~,3, T3,4, T4,1) satisfy: 1. T1,2 and T2,3 are meromorphic on an upper half plane. 2. :/73,4 and T4,1 are meromorphic on a lower half plane. 3. Ti,2T2,3 and T3,4T4,i are meromorphic on all of C and their product is 1.

Then there is a pair ( A , !5) with the same image in H i ( S ) as the cocycle, such that A has coefficients in C(z).

1.1.8. ANALYTIC EQUIVALENCE CLASSES

143

Proof." With the notation of the proof of the last theorem one has that Ti,i+i = Fi-lFi+i. The matrices F1, F2, F3, F4 are defined on a right domain, an upper half' plane, a left domain and a lower half plane. The f'/ are the a s y m p t o t i c lifts on the sectors Si of some /5 E Gl(n,C((z-1)) ). The expression F~iF3 is defined on an upper half plane and on a lower half plane, and is equal to TinT93 and (T3,4T4,1) -1 on those half planes. The third condition states that there is an invertible meromorphie matrix defined on all of C which has F-1F3 as restriction on an upper half plane and a lower half plane. Therefore F1 and /:3 are meromorphie outside a compact subset, of C. The matrix F1 is invertible meromorphic on r < 1:1 < ,x,. We consider the open covering { C , { z I r < Iz} < oo}] of the complex projective line p i ( C ) . Using Corollary 12.8, one concludes that Fl : C D with C and D invertible meromorphic matrices on {z I r < I~l _< oo} and C respectively. In particular, there is an invertible meromorphic matrix B at oc such that BF1 is meromorphic on all of C. The {BFi} are asymptotic lifts on the sectors Si of B F . The coeyele associated with the {BFi} is again (Tl,e, T2,a, 73,4, T
We will show that the conditions in Corollary 11.13 are also existence of a pair (A, F) with the properties stated there.

7~ecessary for the

Let the pair ( : t , F ) be given. The lifts Fi of F have the properties: Fz is meromorphic on all of C, F~ is meromorphic on an upper half plane, F3 is meromorphic on all of C and /:4 is meromorphic on some lower half plane. Hence F~lF2 and F 2 1 F 3 are meromorphic o n an upper half plane. Furthermore F . j i F 4 and F[iF1 are meromorphic on a lower half plane. T h e products (F~IF2)(F,T1Fa) and (F]iF4)(F4-1Fi) are equal t,o F1-1/523 and l < ] l F i and so the third condition in Corollary 11.13 is also necessary. A Problem. Let 9(z + 1) = A 9(z) be a difference equation, say over koo, which has the canonical form 9(z + 1) = A c y(z) over k s . Suppose that an invertible /~' E Gl(n, ~'oo) is given with F(z+l)-lAF(z) = A c. If the equation y ( z + l ) = A ~ #(z) is semi-regular, or more generally satisfies the conditions of L e m m a 8.1, then there are unique asymptotic lifts of F on large sectors. W i t h those lifts and t,he k u o w b d g e of the difference Galois group of the equation #(z + l) = A C y(z),

C H A P T E R 11. W I L D D I F F E R E N C E E Q U A T I O N S

144

which is the formal difference Galois group of the equation y(z + 1) = A one can determine the difference Galois group of the equation 9(z + 1) = A For mild equations the same m e t h o d works if one uses multisummation. In cases, one has produced a canonical choice, i.e. functorial and c o m m u t i n g tensor products, of a cocycle representing the image of the pair (A, F) in H*

y(z), y(z). both with (A~).

In the general case it seems that there is no canonical choice. Is there another m e t h o d to calculate from the canonical equation y(2 + 1) = A ~ y(z) and the image of the pair (A, F) in t t l ( A ~) the difference Galois group of the equation y(z + 1) = A y(z) ?

11.9

An example

The following example illustrates methods and results from the last chapters. Consider the nfild difference equation

9(z+l)=cy(z)+z

-1 with c E R a n d

c > 1.

There is a unique formal solution ~). It, has the f o r m ~nc"Z=l a r t z is to find and compare solutions on sectors which are lifts of Y.

-rz

.

Our aim

W i t h the elementary method of the proof of L e m m a 8.1 one can make a guess for an asymptotic lift vx of Y, namely " :=-

C-n- 1

s Tu

rzm0

It is easily seen that this sum converges locally uniformly on C and is a meromorphic function on C. The function has simple poles at the points 0, - 1 , - 2 , - 3 , . . . . Further vl(z + 1) = cvl(z) + z -1 On the sector (-rr, Tr) the function Vl has asymptotic expansion/). We recall the precise meaning of this: For every ~ > 0 and every N > 0 there are positive constants C, R, depending on c, such that for z with [zl _> R and arg(z) E ( - r r + e, 7r - () the inequality N-1

Ivl(z)-

a.z-' l < Ol21 - N holds. 7;=i

One can verify this by using the expansion c7- ~- -~I __ - Z - i ~ k > 0 C-r~--i ( - n ) kz-k" The function vl is not meromorphie at oo since it has poles at the points 0,-1,-2,-3,.... Therefore 9 is a divergent power series and has explicitly

11.8. A N A L Y T I C

EQUIVALENCE

CLASSES

145

the form co

kTz -- 1

= E(E n=l

-o k>O

We are now looking for a s y m p t o t i c lifts of/) in a sector S at oc which c o n t a i n s the direction 7r. Suppose t h a t there is an a s y m p t o t i c lift v on a sector S of the form ( - e + -}, ~ + Q, with some abuse of n o t a t i o n and with ~ > 0 and small. (We note t h a t m u l t i s u m m a t i o n or the m e t h o d s of this chapter, g u a r a n t e e s an a s y m p t o t i c lift on a smaller sector). T h e e q u a t i o n for v shows t h a t v is m e r o m o r p h i c on C and that the poles of v are simple and form a subset of Z. T h e n v - vl = e ~l~ where log(c) is a positive real n u m b e r , u := e 2~i~ and h is a m e r o m o r p h i c f u n c t i o n of u C C*. On the set C* the f u n c t i o n h has only a simple pole at u = 1. T h u s h(u) = cst + ~ .ec~. . . . h~u ~, with cst some c o n s t a n t and the infinite s u m converging for u E C*. We are going to use t h a t v - vl has a s y m p t o t i c expansion 0 for every direction 4~ with 8 in the u n i o n of the two sectors ( ~ - e , ~ - ) 2 a n d ( 7r5, }_ + e ) For the first sector we write

for the Laurent expansion of h at u = 0. T h e a s y m p t o t i c expansion of k~u,~e, log(c) is 0 in the first sector. This implies t h a t every term k,u'~e ~ log(~) has a s y m p t o t i c expansion 0 in the first sector. Take O in the first sector a n d write z = re ir with r > 0. If k,~ ~ 0 then the limit for r --+ oo of the expression lu~e z l~ I = exp(r(-27rnsin(r + cos(C)log(c)) ) is zero. This is equivalent hog(c) c o s ( c ) with n > ~-~ sin(4' For e small enough this is equivalent with n > 0. We conclude t h a t k,~ = 0 for n < 0. T h u s h0 = cst and h,~ = 0 for n < 0. T h e same calculations, but now with r in the sector ( L a u r e n t expansion

l'~u'~ "= cst E u - ~

+ E

7r

~,

Jr

-~ + e) and the

h~u~ ~ h at ~

lead to 1~ = 0 for n > 0. Ill other words h,~ = 0 for n > 0. C o m b i n i n g the results above one concludes t h a t v = vl. However Vl does not have /) as a s y m p t o t i c expansion in the direction 7r because of the presence of the poles at 0,-1,-2,-3 . . . . . This c o n t r a d i c t i o n shows that:

There is no asymptotic lift of y on a sector of the form ( - ~ + ~, ~- + Q. Even stronger, there is no asymptotzc lift on the closed sector [~,

146

CHAPTER

11. W I L D D I F F E R E N C E E Q U A T I O N S

We make a small variation on the example above. Consider the equation

() Y~ 92

(z+l)=

(

c

0

0

c- 1

)() y~

(:)+

~-~

Y2

This matrix equation has a unique formal solution. From the calculations above one concludes t h a t this formal solution has no asymptotic lift on either one of the two closed sectors [ - 2~, s and [5~, ~-]" It is interesting to compare this with the statements of G.D. Birkhoff concerning asymptotic lifts, as given in [9] and [12]. In [12] it is stated t h a t a difference equation in matrix form (a mild one over C(z) in our terminology) has two principal matrix solutions, k+ and }%, having asymptotic expansions on the sectors ( - ~ - ~, 7 + ~) and (7r - ~, 3~Y + ~) This is apparently not the case for our example above. The two principal solution matrices Y+, Y_ are also present in [9]. It seems t h a t Birkhoff only claims that the two matrices have the correct a s y m p t o t i c behavior on a right half plane and on a left half plane. The two principal solutions have an additional condition on the location of the poles and are then unique. T h e example above forces us to the interpretation of "half plane" as "open half plane". This has the disadvantage that Birkhoff's "connection matrix" y ; - 1 y + does not have an asymptotic expansion 1 on suitable subsets of C. In particular one CalmOt conclude that this connection matrix has coefficients in C(u) with u = e s'~i~, if the original equation has coefficients in C ( z ) . However, Birkhoff's results are valid for a more restricted class of difference equations. Moreover, the condition on the location of the poles of the two principal matrix solutions seems to play an i m p o r t a n t role in his work. We continue the analysis of the example y ( z + 1) = ey(z) + z -1 using the method o f m u l t i s u m m a t i o n (see Section 10.5 and T h e o r e m 11.1). T h e singular directions for the m u l t i s u m m a t i o n are =t=2 and the directions r where e zO~ (with n E Z) has locally maximal decrease for z = re ir and r -~ +oo. This condition on r is equivalent with: the function log(c) c o s ( r sin(C) has a negative nainimum at r Clearly r = 7r. In general, r is the solution in the interval (~3~) of the equation a r e t g ( r = -2rrn. 2'2 All the singular directions have level 1. The limit of r for n -+ + c o is ~ and the other limit is ~-. For a direction d which is not singular, the multisum of ~) (which is here 1-summation or the Borel-1 sum) in the direction d exists and is an asymptotic lift of ~) on a sector (d - 2 - 6, d + ~ + 5) for some positive & The m u l t i s u m for the directions d in an interval of directions (a, b), which does not contain a

11.8. A N A L Y T I C EQUIVALENCE CLASSES

147

singular direction, is independent of d. We apply this to the direction d = 0 and we find that the m u l t i s u m in the direction 0 has the correct a s y m p t o t i c behavior on the sector (-r~, ~). Of course this m u l t i s u m is vl. T h e interval (r rr) does not contain a singular direction. T h e corresponding multisum, say v2 has the correct a s y m p t o t i c behavior on the sector (r - ~, a~) 2 ' Another interesting multisum v3 of 9, with respect to the interval of directions (rr, r has the correct a s y m p t o t i c behavior in the sector (~r + ~-) ' 2 ~ 2 ' More generally, we can take two singular directions a, b, such that the interval (a, b) does not contain any singular direction. T h e m u l t i s u m in the direction d, with d E (a,b), does not depend on d. This m u l t i s u m is an a s y m p t o t i c lift of/) and has the required a s y m p t o t i c expansion on the sector (a - ~, b + ~). In our case this is a m a x i m a l sector for which there exists an a s y m p t o t i c lift of/). We now propose to use the method of this chapter for the study of the a s y m p totic lifts. T h e results imply that for every formal solution of a difference equation and any direction d there is an a s y m p t o t i c lift on a small sector ( d - a , d + a ) , with a > 0, around d. In order to find out whether the formal solution has also an a s y m p t o t i c lift on a given ( m a y b e large) sector, one is led to an investigation of the first cohomology group of the sheaf of solutions of the homogeneous equation which have a s y m p t o t i c expansion zero. We will explain this m e t h o d in detail for our example. Let s denote the circle of the directions at oc. On ,5'1 one considers the sheaf of complex vector spaces 8, defined as S(a, b) is the set of the solutions of the equation y(z + 1) = cy(z) which have a s y m p t o t i c expansion zero on the sector (a, b). Consider the problem:

Does y have an asymptotic lift on the sector (a, b)? There is an open covering {&} of (a, b) by intervals oct, for which there are a s y m p t o t i c lifts Yi. T h e s e t {Yi - Yj} is a 1-cocycle for the sheaf 8 with respect to the covering {S~}. If the image of this 1-cocycle is 0 in H 1 ((a, b), 8) then one concludes that there is an a s y m p t o t i c lift of 9 on the sector (a, b). In the following we investigate the sheaf N and the sectors (a, b) for which the cohomology groups H l ( ( a , b), S) are 0. Every element of S(a, b) has the form e z l~ where h is a m e r o m o r p h i c function of u = e 2~iz. If (a,b) is contained in (0,rr) then h is holomorphic on 0 < lul < a for some a > 0. The function h has a Laurent expansion ~,~=-oo h,~u'~ at u = 0. T h e interval (a, b) C (0, rr) determines which terms are

CHAPTER 11. WILD DIFFERENCE EQUATIONS

148

present in this Laurent series. More precisely, the term u '~ can have a non zero coefficient if and only if the expression -27rn sin(O) + cos(O)log(c) is negative for every O r (a, b). Using this one call show that 9 S ( a , b ) = S ( a , Tr) for 0 < a < b < 9

7r.

S(0, 7r) = 0.

9 $(d, Tr) C S(e, 7r) for 0 < d < e < 7r. The conclusion is that. the sheaf S has trivial cohomology on any subset of the sector (0, ~r). For similar reasons, the sheaf S has trivial cohomology on any subset of the sector (-7r, 0). It. is rather clear that 8 ( a , b ) = 0 if 0 E (a,b). For the calculation of the cohomology of S on the sector (-Tr, 7r) it suffices to consider the open covering {(-Tr, 0), ( - ~ , 6), (0, 7r)}, with 6 > 0, since the cohomology on (-Tr, 0) and (0, 7r) is trivial. T h e (~ech cohomology of S with respect to this covering has clearly the p r o p e r t y H 1 = 0. It follows that H l ( ( - T r , Tr),$) = 0. From this we conclude that ~) has an a s y m p t o t i c lift on the sector (-Tr, lr). This lift is unique since 8 ( - 7 r , lr) = 0 and therefore equal to Vl. We conclude t h a t the sheaf S has trivial H 1 for every open subset, of (-Tr, 7r). One can express this as: "all the 1-cohomology of' the sheaf S is concentrated in the direction 7r". One can also find the a s y m p t o t i c lift v: be a calculation of cohomology. Indeed, by P~emarks 11.9 there is an a s y m p t o t i c lift v of ~) on the sector ( ~ , 2 ' Take (~ > 0 and small. T h e interval (r - 5, ~-) has a covering by (~, T3 ) and (~ - d, 7r). T h e H 1 of 8 with respect to this-covering can be seen to be zero. This proves the existence of v2. More generMly, let a, b be singular directions, such that. (a, b) does not conthin any singular direction. T h e n one call show, using the description above of the sheaf S, that Hl((a - ~,b + ~ ) , S ) = 0. This proves the existence of an a s y m p t o t i c lift on the sector ( a - ~ , b + ~). Since also H ~ ~,b+ r = 0, this a s y m p t o t i c lift is unique and coincides with the multisum with respect to the directions in (a, b).

Chapter 12

q-difference equations 12.1

Formal aspects

We start by analyzing the formal aspect of q-difference equations. We suppose that the number q E C is not zero and not a root of unity. A logarithm 2rrir of q is fixed. On the fields C(z), k ~ := C({z-Z}), ko+ := C ( ( z - t ) ) , k0 := C({z}) and k0 := C((z)) there is a natural action of r given by r = qz. Let K denote the union of the fields C(z 1/~) for m = 1 , 2 , . . . . The symbols z x with A E Q are chosen such that zXlz ~ = z x~+~'~. On the field K one extends the action of r by r ~) = e2~iTXzX. On the algebraic closures of ko, ]r k~o and k ~ the extension of the action of r is defined by the same formula. Let k be any of the fields above. A q-difference module over k is a pair (M, ~) where M is a finite dimensional vector space M over k and (I) : M --+ M is an invertible C-linear map such that q)(fm) = r for any f E k and m E M. The next step is to find "normal forms" for q-difference modules. Let 7~ = 7)oo denote the algebraic closure of ]r This is the field of formal Puiseux series in the variable z -1. The group of equivalence classes of one dimensional modules is equal to P ~ / U where U is the subgroup consisting of the elements ~ with f E 79~. This group U turns out to be {e;~iC"(1 + d)l ~ ~ Q, d ~ z-1/'nC[[z-1/m]] for some m _> 1.} The group P ~ / U is isomorphic with

{,?12, ~ Q} x C+/{e')"~"~' I ~ E Q}.

CHAPTER 12. q-DIFFERENCE EQUATIONS

150

It is not difficult to show t h a t every module over 7)oo is isomorphic to a direct s u m ~ x , ~ E(zxc) | Mx,~ where E(zac) denotes the one dimensional module I)~e~,~ with ~(ea,~) = zacex,~ and where each Mx,~ has a basis such that the matrix of 9 with respect to this basis is a constant unipotent matrix. This representation is not unique. Indeed, E(e 2~i~') ~- E(1) for every rational #. If one fixes a set of representatives S in C* of C * / { e 2 ~ I # ~ Q} then every q-difference module over 7)o0 is isomorphic to a unique direct sum Y~xeq,~es E(zxc) | Max. We want now to construct a universal Picard-Vessiot ring over ]%0 for the q-difference equations over this field. T h e difference ring R := C(z)[{e(zXc)}xeq,~ec 9 ,1] is defined by the following relations: 9

e(z~,cl)e(z~c~) = e(zXl+x~c~c~).

9

e(1) = 1 and e(q) =z.

9

r

= ~(d~i'~)c~(z~c).

9 r

Taking the tensor product with koo over C(z) gives a similar ring

We note that Poo embeds in/~oo in a canonical way be sending z x to e(e~irX). It is not difficult to show t h a t the last ring is the universal Picard-Vessiot ring for q-difference equations over ]%0. This means: 1. Roo has no proper r 2. T h e set of r

ideals. elements o f / ~

is C.

3. Every q-difference equation over ko0 has a f u n d a m e n t a l m a t r i x with coefficients i n / ~ . 4. No proper difference subring has the properties (1) and (3). We note that /~oo is also the universal Picard-Vessiot ring for the q-difference equations over 7)oo. T h e group of all C(z)-linear a u t o m o r p h i s m s which c o m m u t e with r will be denoted by described by a triple (h, s, a) as follows: h : C* --+ C* is a h o m o m o r p h i s m with h(q) = and a is a constant. T h e action of a is given

of R = C(z)[{e(z~c)}xcQ,cec 9, 1], G. Any element c~ E G can be 1, s : Q --+ C* is a h o m o m o r p h i s m by the formulas

(r(e(z~c)) = s(A)h(c)e(h(e2'~i~))e(z~c) and or(l) = l + a.

12.1. FORMAL ASPECTS

15 ]

The group G can be given a natural structure as affine group scheme over C. This structure can be defined as follows. Choose an integer n > 1 and a finitely generated subgroup C C C*, which contains the element e ~ir/n. Let R,~.c denote the subring C(z)[{e(zAe)}Ael/nZ,ceC, I] of R. The ring P~,c is invariant under r and also invariant under each cr C G. Let G,~,c denote the group of the automorphisms of R,~,c which conmmte with the action of r Then G~,c is in an obvious way a linear algebraic group over G. The restriction homomorphism G --+ G~,,c is surjective. Further, R is the filtered union of the }~,~,c- Therefore G is the projective limit of the groups G,~.c. This makes G into an atIine group scheme over C. The subgroup G O of G consists of the r such that the corresponding triple (h, s, a) has the property that the subgroup

of C* lies in the kernel ofh. This group is in a similar way the projective limit of groups G~, c. One easily verifies that each G ~ is commutative and connected. In fact G ~ is a product of a torus and the additive group G~. Further G ~ is the component of the identity of the group Gn,c. Special elements in the group G, which will be used later, are 7 and (~ defined by: 7 has s = 1, a = 0 and hi : C* -+ #oo C C* a h o m o m o r p h i s m s u c h that

hl(e 2~i~) = e 27ri'x for all A C Q and hi maps poo to 1. Here poo denotes the group of the roots of unity in C*. The definition of 7 is chosen such that 7e(z x) = e(e2~i~z ~) and 7e(e 2~i~) = e(e 2~i~) for rational A. 6 has s = 1 and a = 0 and h2 : C* --+ # ~ C C* is a homomorphism which is the identity on ;zoo and maps all e 2~rir)' with A E Q to 1. The a is chosen such that 6e(z )') = e(z ~) and (~e(e2~i)') = e~iAe(e 2~i:~) for rational A. The elemer~ts 7 and ~ do not commute. The commutator c : = ([7(~-17 - 1 has the properties:

ce(c) = e(c), ce(z )') = e~i)'e(z)~), e E G ~ e commutes with 6 and 7. An analysis of the actions of 7 and 5 shows that the only relations in the group generated by "~ and ~ are c7 = 7c and cd = &. This implies that the group generated by 7 and 5 is isomorphic to the Heisenberg group, i.e. the group

{

1 a b) 0 1 c 0 0 1

]a,b, c E Z } .

C H A P T E R 12. q - D I F F E R E N C E E Q U A T I O N S

152

The group G ~ is a normal subgroup of G. The group G / G ~ can be identified with the homomorphisms of the torsion subgroup T o t s of C*/{q~l 'n E Z} to C*. The group T o t s is isomorphic to the product of two copies of Q / Z . The images of ~, and 5 in G / G ~ are "topological generators" of this group. This has the following meaning: For every finitely generated subgroup K of Tors, the group of homomorphisms of K to C* is generated by the images of 7 and 5. This statement follows at once from the definitions of 7 and 5. The q-difference module M over k.~ is called regular singular if the fundamental matrix for M does not involve the terms e(z x) with A ~ 0. There are several ways to reformulate that property. The following statements on M are equivalent: 1. M is regular singular. 2. There is a complex vector space V C M such that V is invariant under the action of 9 and such that the canonical map k ~ ~ V -+ M is an isomorphism. 3. Tile module M has a matrix representation y(qz) = A y(z) with a constant invertible matrix A. The [nodule M over koo is called regular if M is trivial, i.e. M ~ k~o with the ordinary action of 05. The group of the automorphisms of Roo which are the identity on k ~ and commute with r is identical with the group G that we have just described. For a difference module M over koo one defines coco(M) = ker(r - 1 , / ~ ,g~ M). This is the vector space of "the solutions of the equation". The group G acts on ~ooo(M). The image of this group is the difference Galois group G of M. The structure of the group scheme G implies the following results: P r o p o s i t i o n 12.1 Let G be the difference Galois group of a q-difference module M over koo. Then G ~ the component of the identity of G, is either a torus or the product of a torus and the additive group G~. The group G / G ~ is commutative and is generated by at most two elements. Suppose that M is regular singular. Then G is commutative. The subgroup H C G, generated by the images of'y and 5, is a finite commutative subgroup whwh maps surjectively to G / G ~ For q-difference modules over ]e0 the situation is analogous. P r o p o s i t i o n 12.2 Let G denote the difference Galois group G of a q-difference module M over C(z).

i. The group G / G ~ is commutative and is generated by at most two elements.

12.2. A N A L Y T I C

153

PROPERTIES

2. Suppose that koo | M or ]% | M is regular singular. There is a finite commutative subgroup H, with at most two generators, such that the map H --+ G / G ~ is suuective. 3. Suppose that ] ~ | M or ]% @ M is regular then the difference Galois group of M is connected.

Proof." (1) The embedding C(z) C ]%0 induces an inclusion of the difference Galois group Goo of ]%o @ M into G. Let G' denote the subgroup of G generated by G ~ and the images of 3' and (~. Since G ~ C G ~ and because Goo is generated by G ~ and the images of 7 and (~, one has Goo C G'. T h e Picard-Vessiot ring of M is denoted by P V . One can see P V as a subring of the Picard-Vessiot ring P V ' of the module ~:oo | M . For the invariants one has the following inclusion P V a' C ( P V ' ) a ~ = ]%o. Since P V a' is a finite extension of C ( z ) one concludes that P V a' is a field. T h e existence of a C-action on P V a' implies that P V c' = c ( z l / m ) for some m > 1. From P V a' C koo it follows t h a t P V a' = C(z). This implies t h a t G' = - G . Hence G / G ~ is generated by the images of "y and 6. T h e c o m m u t a t o r of the images of 7 and 6 belongs to G ~ C G ~ This proves the statement. (2) If koo | M is regular singular then the images of 3' and (~ in Goo C G generate a finite c o m m u t a t i v e subgroup H of Goo C G. T h e m a p H -+ G / G ~ is surjective. (3) If )%o @ M is regular then Goo = 1 and the group H of (2) is the trivial group. This implies that G is connected. |

12.2

Analytic properties

We start the analytic part by considering q-difference equations over k0. T h e case k ~ is of course quite the same. The following example shows that we have to assume that Iql < 1 (or Iq] > 1) in order to obtain a reasonable analytic theory. Example

12.3

The equation y(qz) = (1 + z ) y ( z ) over ko

T h e coefficients of the unique formal solution y = ~,~>0 a~z'~ with a0 = 1 are given by the recurrence relation (qn _ 1)a,~ = a,~-l. Hence a,~ = (q'~ - 1 ) - 1 . . . (q - 1) -1. Suppose that Iql = 1 and so r E R \ Q. One has ]1 - q~l 0~ = 2(1 - cos(2rrnr)). If the formal series y is convergent then there is a positive r with 1 - cos(2rrnr) > r2'~/2 for all n _> 1. This implies r" for every integer m i n f l n r + Z I _> r"~ for all n _> 1. In other words I v + 71 > -h(and also for every n _> 1). Liouville numbers can be rapidly a p p r o x i m a t e d by rational numbers. If r is a Liouville number then the divergence of the series is m

_

154

CHAPTER

12. q - D I F F E R E N C E E Q L \ 4 T I O N S

"'chaotic" in the sense that there is no hope that the formal series is summable. For real algebraic, non rational r one has the famous Roth's inequality

17-+

n

>- -

n 2+e

with a positive constant c(r, e) depending on r and e. This inequality assures that the series y is convergent. These are (as far as we know) the only concrete examples of real numbers which cannot be too rapidly approximated by rational numbers. One knows however that this subset of the real numbers has full measure. If Iql--fi 1 then the series y is evidently convergent. In the sequel we will assume that

12.2.1

Iql

< 1 or Ira(r) > O.

Regular singular equations over ko

By regular singular equation M over ko we mean that ]r Qko M is regular singular. One easily sees that this is again equivalent to: M has a matrix representation y(qz) = A y(z) with A E Gl(n, C{z}). Let y(qz) = A y(z) be difference equation in matrix form with A = A0 + A ~ z + . . . C Gl(n, C{z}). Let a a , . . . , c ~ r denote the eigenvalues of A0. One may further suppose that the matrix A0 is in upper triangular form and that the first diagonal entry of A0 is ~1. Let F denote the diagonal matrix with diagonal entries z -1, 1 , . . . , 1. The transformed matrix A := F ( q z ) - l A F ( z ) is again in Gl(n, C{z}) and the eigenvalues of A0 are qo~1,c~2,..., a~. A similar process, which changes the eigenvalue c~ into q - l ~ r , can be applied to A. Repeating this process one finally finds an equation y(qz) = B y(z) which is equivalent to y(qz) = A y(z) such that the eigenvalues r of B0 satisfy Lql < I~yl _< 1. The next result shows that regular singular equations are rather simple and do not give rise to problems about asymptotic expansions. L e m m a 12.4 Let A = Ao + A l l + ... be an invertible matrix with coefficients in C{z}. Suppose that eigenvalues a l , . . . , c ~ r of Ao have the property that Iql < I(~jl <_ 1. The unique formal matrix F = 1 + F l z + F~z 2 + . . . with F ( q z ) - l A F ( z ) = Ao is convergent. P r o o f : For the constant matrices F~ there is the following recurrence relation q'~F,~Ao - AoF,~ = A1F,~-I + ... + A,~-I + A,~.

12.2. A N A L Y T I C

PROPERTIES

155

The m a p L,~ : t? ~-4 q n B A o - A o B , n > 1, from the vector space of the matrices to itself is bijective. Indeed, if L~ (B) = 0 then A o B A o I = q'~B. The eigenvalues of the m a p B ~ A o B A o 1 are the c~iaj -1 . Hence B = 0. This shows the existence and uniqueness of the formal matrix F. Since IqP tends to 0, there is a constant e such that the norms of the inverse maps L~ 1 is bounded by c. Using this one easily proves t h a t F is convergent. | As a consequence one obtains that every regular singular difference equation over k0 is equivalent with an equation y(qz) = A y(z) where A is a constant invertible matrix. One can moreover specify that the eigenvalues c of A satisfy Iql < Icl _< 1. W i t h this specification, the constant matrix A is unique up to conjugation in Gl(n, C). A translation of the above for regular singular modules M is the following: Let wo(M) = k e r ( q 5 - 1,ko[{e(c)}r

| M).

Then ~ 0 ( M ) is a finite dimensional vector space over C and the canonical m a p

o

o(M)

Oko M

is an isomorphism. An a u t o m o r p h i s m c, of k o [ { e ( c ) } c e c . , l] which is the identity on k0 and commutes with r is determined by a h o m o m o r p h i s m h : C* --+ C* such that h(q) = 1 and an element a E C. The action of ~ is ~r(e(c)) = h(c)e(c) for all c C C* and ~(l) = l + a. This group is commutative and has the structure of an affine group scheme over C. As before, the group is topologically generated by two special elements ~/and ~ and a connected subgroup corresponding to the the h o m o m o r p h i s m s h which are trivial on

Consequently, the difference Galois group G of any regular singular equation over k0 has the properties: 9 G ~ is a torus or the product of a torus and the additive group Ga. 9 There is a finite c o m m u t a t i v e subgroup H C G with at most two generators such that H -+ G / G ~ is surjective. 9 G is commutative. This finishes the theory of regular singular q-difference equations over the fields k0 and k ~ .

Remarks

12.5 More general equations over ko

CHAPTER i2. q-DIFFERENCE EQUATIONS

156

(1) We give an example to show that formal solutions are in general not nmltisummable. Consider the equation

y(qz)=

(21) 0 1

y(z).

There is a unique formal matrix ( 01

()with

F(qz)-lAF(z):

( 0z O 1) '

n2q-n

The element f turns out to be f = Y ~ > 0 q ~ z ~. This series is divergent and not multisummable since for a mulGsummable series y~a,~z '~ there is an inequality of the form la~l _< ca(n!) c2 for positive constants c~, c~. The classical methods for the interpretation of the formal series f do not work. It is possible that a quite different summation method can be applied in order to give the formal f some meaning. See also [52] (2) Let us suppose for an instant that Iql = 1 and that r G R cannot be too well approximated by rational numbers. In that case one can show that every difference equation over k0 is equivalent (over the algebraic closure of k0) with an equation y(qz) = A y(z), where A is a direct sum of blocks of the form z;~'Ai with rational/~i and constant invertible matrices Ai. This means that the analytic theory of equations over k0 coincides with the formal theory over J%. This looks quite interesting. The disadvantage of the choice of a q with Iql = 1 becomes apparent if one studies q-difference equation y(qz) = A y(z) over C(z). Suppose for simplicity that A(0) = A(oo) = 1. The equation has a fundamental matrix Fo E Gl(n, C{z}) at 0 and also a holomorphic fundamental matrix Foo at oo. The fundamental matrix F0 is defined in some neighborhood of 0. The equation Fo(qz) = A(z)Fo(z) cannot be used to extend F0 to the entire complex plane since Iql = 1. One is not able to compare the two fundamental matrices and no information about the difference Galois group of the equation over C(z) is obtained. This is the second reason why we will take a q with Iql < 1 in the sequel.

12.2.2

Equations

over

C(z)

We continue with the investigation of certain difference modules M over C(z), namely the regular singular ones. A difference module M is called regular singular if both k0 | M and koo | M are regular singular. The module M is called regular if k0 | M and koo | M are trivial difference modules. We start with an example which indicates the general method that we will develop. ' ' with a E C* E x a m p l e 12.6 y(qz) = (z--a)(z--a-1) (z_l)a ytzj This is a regular order one equation over C(z). It has local solutions Y0 and y ~ at 0 and co. The local solutions are normMized by y0(0) = 1 and

12.3. C O N S T R U C T I O N OF THE C O N N E C T I O N M A P

157

y~: (oc,) = i. The solution Y0 is a merolnorphic function on the entire complex plane. The solution yo, is meromorphic on C * U {ec}. The function f := y o l y ~ is meromorphic on C* and q-invariant. Hence f is a meromorphic function on the elliptic curve E = : C * / q z. One can calculate the poles and the zeros of f as follows. The function y.~ has neither poles nor zeroes in a neighborhood of T h e equation y~(qz) = (~-~(_~;~-~)y~(:) shows that the divisor of y ~ is equal to [q"a] + ~ n~>nl

[q'~a-1] + Z

n~n2

-2[q'~] for certain nl, n2, ,13.

7~3

The function y~- 1 has no zeros or poles in a neighborhood of 0. It follows that the divisor of f on E is [p(a)] -t- [p(a-t)] - 2[#(1)], where p : C* + E is the canonical map. One conclusion is that f is constant if and only if a is an integral power of q. Further f is constant if and only if the equation has a non-trivial solution in C ( : ) . The choice c~ = - 1 leads to a function f with divisor 2[/)(-1)] - 2[p(1)]. Hence for this choice, f is the normalized Weierstrass function on E. One can give explicit expressions for Y0 and y ~ , namely

Yo = r i ((qjz _ a)(qJz - a - x ) ) _ t and y ~ = r I (q-Jz _ a)(q-Jz - a - t ) j_>0

j>_l

Using this product one finds a product formula for f . This fornmla is of course the usual way to write a meromorphic function on E as a product of t.heta functions. Remarks: [n the recent paper [22] the Galois group of a matrix equation 9(qz) = A ( z ) y ( : ) with A(0) = A(,~) = 1 is studied. This is a regular equation. For a matrix equation as above, P.I. Et.ingof defines a connection matrix and shows that this matrix "generates" the difference Galois group of the equation. Our methods are somewhat different and extend to the larger class of regular singular equations.

12.3

Construction

12.3.1

Meronaorphic

of the connection map vector

bundles

We will need sonle less known properties of meromorphic matrices. We start with some definitions. For any COlmected Riemann surface X, we write Ox = (9 and ?~4x = 34 for the sheaves of holomorphic and meromorphic functions on X. A holomorphic vector bundle of rank n is a sheaf of O-modules which is locally isomorphic to (9~. A m.eromorphic vector bundle of rank n on X is a

158

CHAPTER 12, q-DIFFERENCE EQUATIONS

sheaf of .s which is locally isomorphic to 3.4 '~. A holomorphic vector bundle can be described by an element in Hi(X, Gl(n, O)) and a meromorphic bundle by an element of H i ( x , Gl(n, A/I)). The following lemma says that every meromorphic vector bundle comes front a holomorphic vector bundle. We give here essentially the proof of C. Praagman [48]. L e m m a 12.7 The canonical map HI(X, GI(n,(9)) --6 H I ( x , GI(n,M)) is sur)ective. Proof." Let L be a meromorphic vector bundle represented by an element E HI(X, GI(n,M)). This element can be calculated with Cech cohomology. The space X is paraeompact and there are two locally finite open coverings U := {U,:} and V = {V/} of X such that: 1. ~ is represented by elements Ai,j E Gl(n,M(Ui N Uj)). 2. V/ is a relatively compact subset of Ui. The set of points Si,j C ~/; N I~, where Ai,j is not an invertible holomorphic matrix, is finite. Since the covering V is locally finite the set S : = L.Ji,jSi,j is a discrete subset of X. Let ~'~,"= I/} \ S and X* = X \ S. Then {}'}*} is an open covering of X" and the restriction of the 1-cocycle {Ai,j} to ~'}* N V.* J has values in Gl(n, 0). The corresponding holomorphic vector bundle L0 has the property that the natural morphism f : .Ol r Lo ~ L]x. is an isomorphism. (Here Llx* denotes the restriction of L t.o X*.) The holomorphic vector bundle L0 can be extended to X since ,5' is a discrete set. This extension, again called L0, can be made such that. L0 is a subsheaf of L. The morphism f extends then in a unique way to an isomorphism f : M @o L0 --+ L. |

Corollary 12.8 1. Any meromorphic vector bundle on X is trivial. 2. For every T E G I ( n , M ( C * ) ) there are Fo E G l ( n , M ( C ) ) and Fo~ E G I ( n , M ( C * U {oc})) with F~IF~, = T. Proof." 1. The meromorphic vector bundle L comes from some holomorphic bundle L0 on X. If X is an open Riemann surface then every holomorphic vector bundle on X is trivial according to Theorem 30.4 of [24]. If X is compact then L0 "is" an algebraic bundle on the connected projective non singular algebraic curve X. Such a bundle trivializes over the function field of X (Corollary

29.17 of [24]). 2. Using the covering of the projective line P ~ over C by the two sets C and C ' O {~c}, one can view the matrix T as a 1-cocycle for Gl(n, M ) . This 1-cocycle is trivial since HI(X, Gl(7~,M)) is trivial. This proves the existence of F0 and f~. |

12.3. C O N S T R U C T I O N OF T H E C O N N E C T I O N M A P

12.3.2

159

T h e c o n n e c t i o n m a p of a r e g u l a r e q u a t i o n

hi the sequel we will use the following notation: M ( X ) is the field of meromorphic functions on ally connected R.iemaml surface X. Let E denote the elliptic curve C ' / q z (which is isomorphic to C / Z r + Z). We note that. the field of meronlorphic functions .,M (E) coincides with the set. of the 0-invariant elements o f M ( C +) Our aim is to construct a connection map for regular singular modules over C ( : ) and to give its relation with the difference Galois group. We start with regular modules M in order to illustrate the main ideas. Put. ~'0(M) = k e r ( ~ - 1,A4(C) ,~ M) and ~',~:(M) =

/~'er(~-

1,.~I(C ~ U {,.,}) (S:,M).

From the local theory at. 0 it. follows that the canonical map

A4(C) ~2"c,~o(M) -+ M ( C ) ~ c ( : ) M is an isomorpism. A similar result holds for w ~ ( M ) . The two spaces w0(M) and w ~ ( M ) have natural maps to w . ( M ) := k e r ( ~ - 1 , M ( C * ) ~ M) and one can compare their images. F/oughly speaking, this will produce the "'connection matrix". The subset of 0-invariants elenlents in M ( C * ) is equal t,o ,VI(E). The m a p ~z0(M) -4 ~ . ( M ) induces an isomorphism M ( E ) C~c aso(M) --+ ~ , ( M ) . There is a similar isomorphism M ( E ) (9c w ~ ( M ) --4 w . ( M ) . From this one finds an isolnorphism

,%I : M ( E ) : : ~0(M) ~ M(E) '~c ~oo(M). The m a p SM will be called the connection map (or matrix) of M . For any regular module M we associate the triple ( w o ( M ) , w ~ ( M ) , S M ) . We note that, unlike the connection maps defined for ordinary difference equations, this connection m a p relates different vector spaces. Let us introduce a category T r i p l e s of triples. An object of this category is a triple (~.~). I:..~, b'), where ~~), I% are two vector spaces over C with the same finite dimension and where ,S : .'UI(E),~ ~o -+ fl4(E) @ ~"~ is a .~'l(E')-linear isomorphism. A m o r p h i s m f = (f0, f ~ ) from the triple (Vo, Voo, S) to the triple (I'i:{,I'~,,S') is a pail" of C-linear maps J'0 : 1/o --4 V0~ and f ~ 9 t ~ -4 V~ such that S'J\I = f ~ S . The tensor product of the two triples (k0, I ~ , S) and (l,'0', t.~, S') is defined as (1/}~ 4~ 1.0, :' " V~ ,:L~' V .~. . . .q' O S'). W i t h those definitions T r i p l e s is in an obvious way a C-linear tensor category. The forgetful functor q : rIMples -~ V e c t c to tile tensor category V e c t c of the finite dimensional vector spaces over C is a fibre functor. Hence T r i p l e s is a neutral C-linear

C H A P T E R 12, q - D I F F E R E N C E E Q U A T I O N S

160

Tannakian category. The main result on regular q-difference equations over C(z) is the following theorem. Theorem

(a)

12.9

The .tap M ~ (wo(M),woo(M),SM) from the category of regular differ'ence modules over C(z) to the category T r i p l e s is an equivalence of C-linear tensor" categomes. The difference Galois group G of the regular q-difference module M over C(z), seen as a subgroup of Gl(~oo(M)), is the smallest algebraic subgroup which contains S-~11(a)SM(b) for all a, b E E such that SM(a) and SM(b) are invertible matrices.

(c) For every connected linear gTvup G there is a regular q-difference module with difference Galois g~vup G. Proof." (a) The functor ~-, indicated in tile statement of (a), is C-linear and preserves all constructions of linear algebra (in particular tensor p r o d u c t s ) . . P is fully faithful if the map H e m ( l , M) --+Hoin()rl, (a0(M),woo(M), SM)) is bijective. The lefthand side consists of the elements m E M with 0(m) = m. The object Y l is equal to (Ce0,Ceoo,T) with Tee = e o o . Hence the righthand side is equal to the set of pairs (v0,v~) C w0(M) x woo(M) with SMvo = roe. From this description it is clear that the map is a bijection. We are left. with proving that any object (V0, k ~ , T) of 'D'iples is isomorphic to some .TM. Put M0 = ~M(C) G V0 and consider the 0-action on this module defined by 0 f ( z ) ' 2 , v = f(qz) r v. Similarly one defines the difference module Moo = A4(C* U {oc}) ~ ~ . The two q-difference modules are defined above C and C* O {oc.}. The two modules are glued over the open subset C* by the isomorphism A4 ( C* ) g) M0 --+ ~ (C*) <:>Mo~ obtained from T by extending the "scalars" from M ( E ) to A4(C*). The result, is a meromorphic vector bundle on P I ( C ) . The isomorphisln, induced by T, is equivariant with respect to the two actions of 0. Hence we found a meromorphic vector bundle over P l ( C ) together with a 0-action. According to 12.3.1 this vector bundle is trivial. Let M denote the set of meromorphic sections of the bundle. Then M is a vector space over C(e). Any rn C M induces restrictions m0 6 M0 and moo ff Moo. The two elements o(rno) and r coincide above C*. Therefore 0(m0) and 0(moo) glue to an element in M which will be denoted by ~(m). This defines 6 on M. The verification that (wo(M),cooo (M), SM) is isomorphic to (V0,1/%o, T) is straightforward. (b) Consider an object (V0, Voo, T) of T r i p l e s . It generates a fltll tensor subcategory {{(~/}), Voo, T)}} of T r i p l e s . The functor r/: {{(V0, Voo,T)}} -+ Vectc, given by TI(Wo,We~,S ) = We, is a fibre functor. The category is therefore a neutral Tannakian category and is isomorphic with the category of the finite dimensional representations of some linear algebraic group G, in fact a subgroup of

12.3. C O N S T R UCTION OF THE C O N N E C T I O N MA P

161

Aut(V0), which represents the functor Aut | (77). For a, b E E such that. T(a) and T(b) are invertible maps we will produce an element k EAut'a(r])(C) = G(C). By definition (see [20] ) k is a family {.~(wo,w~,s)}, where (W0, Wc~, 2) runs over the set of objects of {{(V0, Voo,T)}}. One defines A(Wo,W~,Sl = S - l ( a ) S ( b ) 9 Since (W0, Woo, S) is obtained from (V0, ~ , T ) by some linear constructions (i.e. tensor products, duals, subquotients) the maps S(a) and S(b) are invertible and the family {)~(Wo,W~,S)} thus defined has the required properties. Let H denote the smallest algebraic subgroup of Aut(V0) which contains all T -~ (a)T(b), with a,b E E, for which T(a) and T(b) are invertible maps 9 Then one finds a functor of tensor categories {{(V0,V.~o,T)}} --+ ReprH, where ReprH is the neutral Tannakian category of the finite dimensional representations of H. It is an exercise to show that this funetor is an isomorphism of tensor categories. This shows that G = H. Using the isomorphism of (a) one finds the result (b). (c) Let a connected linear algebraic group G be given as a subgroup of some Aut(V0). Choose 1/~ := Vo. The field ,M(E) contains a subfield C(t) with transcendental t. According to proposition 3.3 there is some T E Aut(C(t) Q V0) with T(0) = 1 and G is the smallest group with T E G(C(t)). One can see T also an element of Aut(.Ad(E) 7~, V0) and it has the required property. II

R e m a r k s 12.10 Regular equations in matrix form.

If the regular module M happens to have a basis such that the corresponding matrix equation y(qz) = A y(z) satisfies A(0) = A(vc,) = 1 then one has another way of introducing the "connection matrix". (See [22]). The fundamental matrix F0 at 0 is chosen such that F0(0) = 1 and sinrilarly, the fundamental matrix Foo is chosen with Foo(oo) = 1. One easily proves the formulas

Fo = I-[ A ( q n z ) - i = A ( z ) - l A ( q z ) -1

9and

n>O

F~ = H A(q-'~z) = A(q-lz)A(q-'~z)

'

n>O

The matrix T = FolFoo is equal to 1-[,~oo=-ooA(q-nz) 9 Let 8M denote a matrix for SM with respect to some basis of c00(M) and cooo(M). It is not difficult to see that SM = C T D for some constant invertibte matrices C, D. The algebraic subgroup of Gl(n, C) generated by the SM1(a)sM(b) is conjugated to the algebraic subgroup generated by the T-l(a)T(b). The last group is the group appearing in the paper [22]. |

CHAPTER 12, q-DIFFERENCE EQ UATION9

162

12.3.3

The connection map of a regular singular equation

We will now consider a general regtdar singular difference module M over C(z). Let w0(M) denote

k~(~

- 1,M(c)[{~(~))~ec.

,t] ,>~ M).

From the local theory at. 0 we know that the canonical map

M(C)[{e(e)}~ec. ,~] Oc ~0(M) --* M(C)[{e(c)}~ec. ,t] Oc(~) M is an isomorphism. Let Gr~ denote the group of the authomorphisms of C(z)[{e(c)}c~c,, l], which are C(z)-linear and commute with ~. As in 12.1, this is an affine group scheme. It is commutative. In fact G ~ is a quotient, of the affine group scheme G studied in 12.1. The group Gr~ acts on w0(M). One defines w ~ ( M ) in a similar way. It has analogous properties. We consider also the difference ring . ~ := ~4(C~)[{e(c)},l]. The group Grs acts as group of aut.omorphisms of this difference ring. Its subring of 4)-invariant elements will be denoted by ~4~. Since the action of G~.~ commutes with ~ the group Grs also acts o n .A4~. Let w , ( M ) denote

~'e,'(~ - 1,.A4(C')[{~(~)}, Z] C~ M). This is a module over the ring A4~ and has a Grs-action. For f E .A4~, m E w,(M) and (7 E G ~ one has or(fro) = cr(f)(r(m). There is an obvious C-linear and G~-equivariant map from ~0(M) to w,(M). This map extends as a G ~ equivariant map .M~ (5~c w0(M) --+ ,J,(M), where the action of a E G~.~ on the first term is defined by the fornmla ~ ( f (~ m) = ~(f) ~ or(m) for f E ,M~ and m E w0(M). We want to show: L e m m a 12.11 The induced Ad~,-linear and G~-equivariant map

M*. ec ;o(M) --+ ~.(M) is an isomomorphism. Proof." We choose a matrix equation y(qz) = A y(z) with A E (Tl(n,C(z)) representing tile module M. Let Fo E Gl(n, Ad(C)) satisfy Fo(qz)-lAFo(z) = C where (.' is a constant matrix in Jordan normal fornl and eigenvalues cl . . . . , c,,. Write C = C~exp(N) with (-'~s a diagonal matrix with diagonal entries cl . . . . . c,~ and N a nilpotent matrix commuting with Us,. A basis for w0(M) are the columns of the matrix Go := Foe(C~)exp(tN). Here e(6',~) denotes the diagonal matrix with the entries e(el) . . . . . e(c~) on the diagonal. These colunms remain

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unchanged when one considers their image in ~VI, ~,.3M. T h e matrix is invertible over ,,Vt. and so the m a p Ad, (/~,;0(M) --+ M , *~ M is a bijection. Any solution y with coordinates in 3.4, of the equation y(qz) = A y(z) has the form Gov where v is a vector with coordinates in 3 4 ,~. This shows that .M,~ C&c -;0(M) --+ w, (M) is an isomorphism.

There is also an isolnorphisln jk4,~ C>c w ~ ( M ) --+ ua,(M). T h e two resulting isomorphism

fo,fo~ : M.~ 4~c w(M) -+ ~ . ( M ) induce a ,a-4,~-linear and G ~ - e q u i v a r i a n t isomorphism

&v, : ~vff ,.Sc ~0(M) ~ M . ~ ~ c ~ ( M ) , which will be c a l l d the connection map. Before we go further we will need to make the ring ,,k4~ more explicit.

Remarks

The ring.Ad~.

12.12

T h e structure of the ring A4,~ is s o m e w h a t complicated. For the study of its structure we have to introduce some theta functions for the modulus q. Let us define the basic theta function 0 by the formula

o(:) = I-[(~ - v".--~) I-J(1 - q'~:). n > ()

7~ > 0

This expression is a holomorphie function on C* and its divisor is ~,~ez[q'~]. Put 0:~.(z) = 0(x-~z) for any ae C C*. T h e divisor of 0, is ~-~,~6z[ 32 q 17 ]" T h e f u n d a m e n t a l formula is

o.(qz) =

X

ox(~).

--qz

The function e(c), := ~ has the p r o p e r t y e(c),(qz) = ce(c),(z). Therefore e(c)jle(c) is a #5-invariant element o f . M , . It is not difficult to show that M , ~ is equal to the ring : M ( E ) [ { ~ } ] . We note that e(1), = 1, e(qc), = ze(c), and

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that in general e(cl).e(cu). ~k e(clc2).. Therefore A d ( E ) [ { ~ ) ] group algebra" of the group C" over the field 3/I(E).

Lemma

is a "twisted

1 2 . 1 3 The ring,g[** has no zero divisors.

Proof." Let A be a finitely generated subgroup of C ~. Then A is a product It,, x A0. The group A0 is free of rank m and #~,. is the group of the n-th roots of unity. The subring A d ( E ) [ ~ ] ~ e A of 3/i** can be written as

./V4(E)[B,X~,X~ 1 (~)-'~

E M(E).

, "

,

X .... X,~ 1] with B =

Since (e0-~)-'~ ~ M ( E )

~(r

e((~).

"

The only relation is B '~ =

for 1 _< ,n < n, one sees t h a t the

equation Y" - (0@_)_,, is irreducible over ~ 4 ( E ) . This implies that . M ( E ) [ B ] is a field and so ~ ( E ) rL ~(~/1 ~JceA

has no zero divisors.

|

This makes the ring ~M~ in some sense more pleasant. It is of course not possible to give an 6-equivariant embedding of :t4~ in :~4(C*). This is a justification for our use of symbols. We shall now give an interpretation of the ring ,M.~ as ring of meromorphic functions on a natural space. Several copies of C and C* have to be introduced. In order to distinguish them we will add a variable to the space. One considers first the sequence of Riemann surfaces Ct --4 C~ -+ E with z = e 2rcirt. Further coverings Ct --+ C~ and C~ -+ C~ are defined by the formulas u = e 2rcit and u = e 2~riv. The action of (~ on the various spaces is indicated by the action of 4) on the variables. The actions are given by 4)(t) = t + 1, 0(u) = u and o(v) = v. The difference ring .a,4, = . M ( C ] ) [ e ( e ) , I ] ~ e c . is mapped to ,~4(Ct)[e(e)]cr in the following way. Choose a Q-linear subspace L of C witil L @ Q = C. Every element c C C* is then written as eOCl with Co E /~oo and cl = c 2 n i a with a C L. The m a p is given by the usual inclusion 3 d ( C ~ ) C ~ ' l ( C t ) and l ~ t and e(c) ~-~ e(co)e 2~iat. It can be verified that A4. --9 A/l(Ct)[e(c)]~r is a r and injective h o m o m o r p h i s m . The ring A/l(Ct)[e(c)]~r can also be written as ,M(Ct)[f(A)]xEQ with f()~) : = e-2~i)'te(e2'~i~). T h e f ( A ) ' s are qS-invariant elements. They have the relations f ( k l ) f ( k : ) = f(A1 + k2) and f ( k ) = u - x if ,~ C Z. The ring of r elements of 2t4(Ct)[e(c)]ce,~ is therefore ,M(C~,)[f(A)]~r In particular, f ( 1 / m ) "~ = f(1) = u -1. Now one uses the m a p Cv --+ C~. This offers an embedding, which is of course cp-equivariant, of 3/l(C~)[f(A)]a~Q into .,~4(C~) given by the usual inclusion e~4(C;'~) C ~gI(Co) and f ( k ) ~+ e -2'~i)'v. T h e final conclusion is:

j~4~. can be embedded into A4(C~,). More precisely .Ad~. embeds into the field O,~>_lA/J(C~)(e2"iv/'~). The connection map has coordinates in this field.

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165

With this terminology one is able to understand the work of G.D. Birkhoff on q-difference equations, in particular, the papers [10] and [13]. In the first paper, the function H : = q ( t 2 - t ) / 2 :- e rrir(t~-t) is introduced as a solution of r = zH. The function z ~ = e 2~irbt is introduced as a solution of the equation r = e'e'~i~bf. Here is of course a slight problem! The qdifference equations studied in the first paper are essentially regular singular. Birkhoff asserts that the connection matrix is an invertible meromorphic matrix on Ct with period 1. T h a t would mean a matrix with coordinates in Ad(C~,). This is true if the roots of unity do not appear in the formal classification of the q-difference equation. In general however this is wrong for the canonical choice of the connection matrix. In the two papers, the formal classification of the equation is given by producing a fundamental matrix with coordinates in M ( C t ) . Here again the use of the expression z b gives rise to some confusion. | We consider now the category G~s-Triples. An object of this category is a triple (V0, b ~ , T) where V0 and Voo are finite dimensional G~-representations of the same dimension and with

V : M~. Oc Vo ~ M~. Oc V~ a .,Vl.e-linear and Grs-equivariant isomorphism. The category G r ~ - T r i p l e s is in an obvious way a neutral Tannakian category with fibre functor

(Vo, ~/~,T) ~ Vo. Theorem

12.14

(a) The functor M ~-~ ( w o ( M ) , w ~ ( M ) , S M ) from the category of the regular singular q-difference modules over C(z) to the category G r s - T r i p l e s is an equivalence of C-linear tensor categories. (b) The ring ]t4~. is considered, as above, as a subring of the field of meromorph~c functions M ( C ~ ) . The difference Galois group G, seen as an algebraic subgroup of Aut(wo(M) ), is the smallest algebraic subgroup which contains the image of Grs and the SMI(a)SM(b) for all a,b E Cv such that SM(a) and SM(b ) are invertible matrices. P r o o f : (a) The proof is rather similar to that of 12.9. We will only give the proof that any object (V0, V ~ , T ) of G r ~ - T r i p l e s is isomorphic to some

(~0 (M), ~ (M), SM). The module Ad(C)[{e(c)}, l] | V0 has an action of r and Grs. The two actions commute. Let M0 denote the set of invariants under the action of Grs. Then M0 is a vector space over M ( C ) with a C-action. Moreover the canonical m~p

M(C)[{e(c)},/] On(c) M0 -- M(C)[{eCc)},l] Gc V0

CHAPTER 12, q-DIFFERENCE EQUATIONS

166

is an isomorphism, which comnmtes with the actions of q5 and G,.~. One defines Moo in a similar way. The G~s and ~b-equivariant map T induces, by taking Grs-invariants, a ~5-equivariant isomorphism 2t4(C*) c~ M0 -+ M ( C * ) ~ M ~ . As in 12.3.1, the modules M0 and Moo glue to a q-difference module M over C(z). One easily verifies that the triple of M is isomorphic to the given triple

( Vo, V~, T). (b) It is not difficult to show (along the lines of the proof of (b) of Theorem 12.9) that the linear algebraic subgroup of Aut(V0) corresponding to an object (Vo, V~o,T) of G ~ - T r i p l e s is the smallest algebraic subgroup containing the image of G~s and the T(a)-tT(bi (for all a,b E C~ such that T(a) and T(b) are invertible matrices). II

12.3.4

Inverse problems

We will make the last theorem more explicit by considering a subclass of the regular singular q-difference equations over C(z). A q-difference module M over C(z) will be called semi-regular of tgpe n if (a) k0 G M is isomorphic to a direct sum of one dimensional modules koe3 with the property ~(ej) = Xjej and A~ C {qm[ m E Z}. (b) A similar condition for koo Q M. The following statements are easily seen to be equivalent: (1) M is semi-regular of type n and of dimension d over C(z). (2) The vector spaces

to(M) := ker(~ - 1,ko[e(e~i/~),e(e~iT/~)] | M) and too(M) := ker(r - 1,koo[e(e2~i/'~),e(e2'~i'/'~)] | M) have dimension d over C. (3) Let y(q~) = AU(~) denote a m a t r i x equation corresponding to M, then the equation is both over k0 and koo equivalent to a matrix equation with a diagonal matrix with entries in the group {A E C*t k '~ E qZ}.

We also introduce r, (M) for a semi-regular q-difference module M by

r,( V ) = ker((~ - 1, A4( C*)[e(e2~i/'~), e(e2~i'/~)] | M).

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167

Let H denote the group of the automorphisms of C(z)[e(e>~i/n), e(e2'~iT/~)] over C(z) which commute with the action of ~. Then H is the product of two cyclic groups of order n. We need to investigate the set K,~ of the qS-invariants of the ring Ad(C*)[e(e>~i/'~), e(e2~'/~)]. As in 12.12 one can verify that

e(e2rril n ] r,o =

e(e2~rir/n ) 1

~(e~"~/") by E1 and E2. They satLet us denote the elements ~~ ( ~ ' / " ) and e(e2~,~/~)" isfy the equations E~ = fl and E; ~ = f 2 , where fl, f2 are certain elements of Ad(E). It follows that A',, is a finite field extension of Ad(E) of degree at most n u. The group H acts on I(~. Put ~n = e2rri/n. For every tupie (jl,j2) with 0 _< jl,J2 < n, there is a unique element r E H such that cre(e 2~i/'~) = (J~e(e 2~i/n) and cre(e 2~i~/~) = (J~e(e2~i~/'~). For this ~ one has ~r(Ei) = ~J'Ei for i = 1, 2. Hence Kn is a Galois extension of.Ad(E) with Galois group H. One can verify that K~ is actually the function field of an elliptic curve F over C. The group H actg on F as the group of the translations over the group Fin] of the points of order n of the elliptic curve F. The curve E is then isomorphic to the quotient F/FIn]. Thus the curve F is actually isomorphic with E. We introduce the neutral tensor category H - 1 M p l e s analogous with the notations before, as the category having objects (V0, V~, T) with: (1) ~89 V~ are finite dimensional vector spaces over C provided with an H-action. The two vector spaces are supposed to have the same dimension. (2) T : Kn | Vo -+ Kn @ Vo~ is a Kn-linear and H-equivariant isomorphism. For a semi-regular q-difference module M of type n there are natural K,~-linear and H-equivariant isomorphisms Kn Q r0(M) --+ r, (M) and K.. ~) r ~ ( M ) --+ r , ( M ) . The resulting "connection map" o

0(M) -+

0

will again be denoted by SM. C o r o l l a r y 12.15

1. The functor M ~-+ (T0(M), too(M), SM) from the category of the semiregular q-difference modules of type n over C(z) to the category H - T r i p l e s is an equivalence of C-linear tensor categories. . The difference Galois group of a semi-regular q-difference module of degree n over C(z) zs the smallest algebraic' sub9roup G of Aut(ro(M)) such that the image of H belongs to G and S M ( a ) - I S M ( b ) G G for all a,b G F such that Sm(a) and SM(b) are invertible maps.

168

C H A P T E R 12, q - D I F F E R E N C E E Q U A T I O N S

Let us study H-triples more closely. The Tannakian group associated with the triple is the group described in the corollary above. The main question is to find out what the possibilities for this Tannakian group are. We make a slight extension of the notion of H-triples by introducing H, K, Ltriples. The K, L are fields, chosen once for all, such that C C K C L. The field L is supposed to be a Galois extension of K with a Galois group H which is a finite commutative group. A triple (Vo,Vo~,T) is called a H, K , L - t r i p l e if p0 : H -+Aut(V0) and po~ : H --+Aut(Voo) are two representations of the same finite dimension; T : L Q V0 --+ L | Vo~ is an L-linear and H-equivariant isomorphism. The Tannakian group of a triple can be seen to be the smallest algebraic subgroup G of Aut(V0) such that: (a) G contains the image of H. (b) For one (or any) linear isomorphism A : Voo --+ V0 the map A T lies in a coset B G ( L ) with B E Aut(V0). Indeed, T is already defined over a finitely generated and H-invariant Csubalgebra OL of L. Then it is easily seen that G is the smallest algebraic subgroup of Aut(V0) which contains the image of H and T ( a ) - l T ( b ) for all maximal ideals a, b of OL. The last statement translates easily into (a) and (b). Let, for the moment, Z denote the smallest algebraic subgroup of Aut(V0) such that condition (b) holds (and thus Z is the smallest algebraic group containing all T ( a ) - Z T ( b ) with the notation above). Then Z is connected and thus Z C G ~ The H-equivariance of T implies that po(h)-ZZpo(h) = Z for every h E H. Then Z has finite index in the group Z ~ generated by Z and po(H). Thus Z' is an algebraic group. It is clear that Z ' satisfies (a) and (b). Hence Z j = G and Z = G ~. A necessary condition for an algebraic subgroup G of Aut(V0) to be the Tannakian group of a triple is therefore: There is a group homomorphism p : H --+ G such that H --+ G --+ G I G ~ is surjective.

The problem is to show that this condition is also sufficient. At the moment we cannot answer this question. The following weaker result holds. T h e o r e m 12.16 Suppose that K / C has transcendence degree one. Let p : H --+ Aut(V) be a finite dimensional complex representation. Let G be an algebraic subgroup of Aut(V) such that: (1) p(H) C G and p : H --~ G -+ G / G ~ is surjective.

12.3. CONSTRUCTION OF THE CONNECTION MAP

169

(2) p(H) lies in N(G~ ~ i.e. the component containing the neutral element of

the normalizer N ( G ~ of G ~ in Aut(V). Then G is the Tannakian group of a H, K, L-trzple.

Proof." We start by proving that any finite commutative subgroup A of a connected algebraic group Z C Aut(V) lies in a (maximal) torus of Z. (This is probably well known, but we could not find a reference for this). It suffices to show that A lies in a Borel subgroup of Z ([29], 19.4). Let A be generated by s elements a l , . . . , a ~ . Let B denote a Borel subgroup of Z containing al and let X denote the projective variety Z / B . The element al acts on X by right multiplication and the set Y of fixed points of this action corresponds to the set of Borel subgroups containing al. Furthermore, Y is a nonempty closed subset of X. By induction there is a torus S containing a2, . . . , a~. Since Y is a projective variety and S is connected and solvable, there is a fixed point for the action of S on Y. This fixed point corresponds to a Borel subgroup containing A. The triple will be (V,V,T) where T E G~ C Aut(L | V) has still to be specified. The group N(G~ ~ is the component, containing 1, of the normalizer N ( G ~ of G O in Aut(V). Let S C N(G~ ~ be a minimal torus which contains p(H). The rank of S is equal to the minimal number of generators of p(H). Let X denote the group of characters of S. One decomposes V as | such that for all s E S and all X, the element s acts on Vx as multiplication by )C(s). The dual of the group H is written as/2/. There are elements ed E L, d E f / such that. L = OaeftKed and h(ed) = d(h)ea for all h E H and all d E /2/. The homomorphism p : H --~ S induces a homomorphism t? : X --+ 1//. We consider E E Aut(L | V) given by: the restriction of E with respect to each Vx is the nmltiplication by eaz. The element E belongs to S(L) and hence also to the normalizer of G~ The Galois group H of L / K acts in a natural way on the elements A E Aut(L | V). The image of A under h E H is written as hA. By construction hE = p(h)E = Ep(h) for every h E H. An element T E G~ is H-equivariant if and only ifT(hL| = (hL| all h E H. (Here hL denotes the ordinary action o f h on L). The last condition can be written as hT = p(h)-lTp(h) for all h E H. This leads to the following translation of the H-equivariance of elements of G~

T E G~

is H-equivariant if and only i f T = E - 1 S E with S E G~

We want to find an S E G~ such that T := E - 1 S E does not lie in B Z ( L ) for any proper algebraic subgroup Z of G ~ and any B E Aut(V). There are elements 91 . . . . ,9~ E G ~ which generate G ~ as an algebraic subgroup of

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170

A u t ( V ) . Choose a transcendental element t E K, the subalgebra C[t] of K and its integral closure OL in L. Take distinct a 0 , a l , . . . , a s E C. Above each maximal ideal (t - a3) of C[t] we choose a maximal ideal bj of OL. We m a y (and do) choose the aj such that E is defined and invertible at each bj. The element S E G~ will be chosen in G ~ Hence S is seen as a rational m a p of p1 + G o. This rational m a p is chosen such t h a t S(a0) = 1 and S ( a j ) = E(bj)gjE(bj) -1 for j = 1 , . . . ,s. Such an S exists since G ~ is a rational variety. The element S has coefficients in a localization C[t]] of C[t]. T h e n T = E - 1 S E has coefficients in some localization (OL)g. The bj are still maximal ideals of (OL)g. By construction T(bo) = 1 and T(bj) = gj for j = 1 , . . . , s . T h e set {T(bo)-lT(by)lj = 1 , . . . , s } generates G ~ as an algebraic group. This shows t h a t T does not lie in any B Z ( L ) , with Z a proper subgroup of G ~ and B E Aut(V). II The next corollary follows from 12.15 and 12.16 for the choice R" = .Ad(E), L = JV4(F) (for suitable F) and H --+ Z a surjective h o m o m o r p h i s m . C o r o l l a r y 1 2 . 1 7 A linear algebraic group G C Gl(n, C) is the difference Galois group of a regular singular q-difference equation over C(z) if (1) G contains a finite commutative subgroup Z which has at most two generators

and which is mapped surjeetively to G / G ~ (2) Z lies in the connected component of the normalizer of G ~ in Gl(n, C). We conjecture t h a t in this corollary condition (2) is superfluous. As an example we will work out the case where the group G ~ is a torus. Example

1 2 . 1 8 G with G ~ a torus

T h e m e t h o d used in this example is a variation on the proof of l e m m a 8.12. T h e character group of the torus G ~ is written as X. There is an H - a c t i o n on X . We claim t h a t it is enough to produce a h o m o m o r p h i s m T : X --4 L* such that: (a) T is H-equivariant. (b) The preimage T - I ( C *) is 0. Let us assume that we have such a T. We m a y identify G ~ with the group of diagonal elements of some Gl(n,C). Let X/ be the character of G O t h a t corresponds to the projection onto the i th diagonal element. Finally, let S = d i a g ( T ( ) ~ l ) , . . . , T ( x n ) ) . Clearly S G G~ An easy calculation shows t h a t conditions (a) and (b) imply t h a t B C G ~ and t h a t S is H-equivariant and t h a t S is not contained in some B Z ( L ) with Z a proper algebraic subgroup of G ~. Let Z[H] denote the group ring of H over Z. This is an example of a free Zmodule with an H-action. In fact any H - m o d u l e X (free and finitely generated

12.3. C O N S T R U C T I O N OF THE CONNECTION M A P

171

over Z) can be e m b e d d e d in Z[H] N for some N _> 1. It suffices to p r o d u c e an element T for the last m o d u l e having the required p r o p e r t i e s . To do this, we will find elements f l , . 9 9 fN E L* such t h a t -] .~e . . .. .. ..... . j,.~ h (fj) ~ mh., C C* for an Y inte g ers m h,~, 4 im Plies t h a t all rn h ,a. are 0. Definiffg T ( E h s u mh,,h,..., 2 h e . mh/vh) = 1-Ih,u,~<~
the order of H . Take an unT h e r e is a r a t i o n a l function fl(aN) = 1. S u p p o s e t h a t has a pole or a zero in some

For the c o n s t r u c t i o n of f2 we take an u n r a m i f i e d p o i n t b C F2 such t h a t f l has no zeroes or poles on the set rr -1 = { b l , . . . , b N } . One takes for f2 a r a t i o n a l function on F2 such t h a t f2 has non zero values on 7 r - l a a n d f2(bl) = 0 a n d f2(bl) . . . . . f2(bN) = 1. Consider a p r o d u c t F : = 1--[h h ( f l ) '~h'~ 1--[hh(f2) "~"'2" If one of the e x p o n e n t s rrth,i is not zero then F has either a pole or a zero in the set r r - l ( a ) U r r - l ( b ) . So the pair f l , f 2 has a l r e a d y the required p r o p e r t y . By i n d u c t i o n one can c o n s t r u c t a t u p l e ( f l , . . - , fn) with the required p r o p e r t y for any n. T h i s finishes the verification of the c o n j e c t u r e above for tori.

Example

12.19

Order one equations over C ( z )

Consider the order one e q u a t i o n y(qz) = a(z)y(z) for any a E C ( z ) * . T h e divisor of a is m0 [ 0 ] - moo [oc] + ~ a j e c" m j [aj]. Write a = ( - z ) "~~co bo with e0 a c o n s t a n t and b0 with b0(0) = 1. S i m i l a r l y a = (-z)'~cooboo with coo a c o n s t a n t and boo with boo(oo) = 1. T h e expression f0 : = I-[j>_obo(qJz) -1 converges on C a n d has the p r o p e r t y fo(qz)-lafo(z) = (-z)'~~

T h e function H(z) : =

1

satisfies H(qz) = - z H ( z ) . We find a solution foH'~~ E 3/l. of our e q u a t i o n . P u t foo : = 1-Ij>l boo(q-Jz). T h i s function converges on C U {oc} a n d gives a second solutioIi-fooHm~e(coo) E /t4.. T h e c o n n e c t i o n m a t r i x , in an e x t e n d e d sense b e c a u s e we have allowed singularities of a at 0 a n d oc, is equal to

foofolHm~176

1) E M~,.

Let us write $ ( a ) for the c o n n e c t i o n m a t r i x of a E C(z)*. T h e m a p a ~-~ 8 ( a ) is m u l t i p l i c a t i v e (as it should be). T h e c o n n e c t i o n m a t r i x of the e q u a t i o n y(qz) = (1 - d-lz)y(z) can e x p l i c i t l y be c a l c u l a t e d and gives the answer :-~te(d-1). An a r b i t r a r y element a E C(z)*

C H A P T E R 12, q-DIFFERENCE EQUATIONS

172

can be written in the form a = c ( - z ) '~ 1--[j(1 - a - f l z ) mj with aj distinct elements of C* and the mj E Z. T h e connection matrix of a is therefore:

H(--aj)mi.H(~-)m~.e(l-Ia;rni). J

J

J

T h e equation corresponding to a is regular if rn = 0, c = 1 , ~ mj = 0 and

I-Ij(-aj) m' = 1. In t h a t case S(a) = I - [ j ( ~ z ) m' is a meromorphic function on E. A n o t h e r interesting equation is y(qz) = y(z) + a with a E C(z). This equation is regular if a has a zero at z = 0 and z = oo. In t h a t case the "connection matrix" is the convergent sum ~ j e z a(qJz). For a = ~ the connection matrix has as a function on E a pole of order 2 at 1 E E. The connection matrix is therefore a linear combination of the Weierstrass function w and 1. In some texts, this sum }--]j ~ is taken as a definition of the Weierstrass function. For a = ~ the connection matrix has (as a function on E) a pole of order 3 at t E E. This connection matrix does not belong to the subfield C(w) of Ad(E).

Example

1 2 . 2 0 Some equations of order two

In this example we investigate the possibilities for q-difference equations over C ( z ) of order two and with difference Galois group Doo, i.e. the a u t o m o r phisms of V = CVl + Cv2 which have determinant 1 and permute the two lines

{Cvl, Cv2}. Let us study first the semi-regular modules. T h e group Doo is generated by the torus {

(01) 1

0

0

c -1

I c E C*} and k :=

, an element of order 4. The semi-regular module will be of type 4

and is given by the following choices: (1) A h o m o m o r p h i s m p : H -+ A u t ( V ) with image the group generated by k. Here, H is the Gatois group of K 4 / K = ]td(E). (2) A c o n n e c t i o n m a p T =

( a 0

0 ) a -1 '

T h e equivariance of a E I,f~ has the explicit meaning: h(a) = a if p(h) is an even power of k and h(a) = a -1 if p(h) is an odd power of k. Let L C K4 denote the quadratic extension of K of the elements of K4, which are fixed under all h E H such t h a t p(h) is an even power of k. T h e non-trivial a u t o m o r p h i s m of L / K is denoted by u. T h e n a E L and ao'(a) = 1. T h e choice (2) is therefore equivalent to the choice of an a in L with ao'a = 1 and a =r 1, - 1 .

12.3. C O N S T R U C T I O N OF THE C O N N E C T I O N M A P

173

Let. M be a general q-difference module over C(z) with difference Galois group Dec. Let P V denot, e the Picard-Vessiot ring of the equation. Then the set of invariants of P V under G,,~ is a quadratic extension of C(z), having an extension of the action of r There are three possibilities for this quadratic ext.ension C(z)[t], namely t = e ( - 1 ) , e(e~ir), e(-e~i'). Iu the first case, the D ~ - t o r s o r associated to M is trivial. This means that M can be represented by a matrix equation 0

y(qz)

=

A

- A -1 ) 0

y(z),

with .4 E C(z). On tile other hand, a difference equation of the type above has a difference Galois group G contained in D ~ . A criterion on A E C(z)* for G = Dec is the following: The above difference equation has Galois group Do~ if and onlg if the equation y(q2z) = ( - A ( z ) A ( q z ) - t ) N y ( z ) has only for N = 0 a non trivial .solution in C ( z ) .

P r o p o s i t i o n 12.21

Proof." Let. (91, Y2) ~ 0 denote a solution of the matrix equation with 91, Y2 in the Picard-Vessiot ring P V of the equation. Then r = - A ( z ) A ( q z ) - i y i (z). Further r = -9192 and so e ( - 1 ) E PV. Hence P V contains two independent solutions 91 and e ( - 1 ) g l of the equation 02(9) = - A ( z ) A ( q z ) - l y . A fundamental nlatrix f~ the matrix equati~

is ( y19-" -e(-1)y2e(-1)Yl) ' with

Ye = A ( q - l z ) C - l Y x . This proves that the Picard-Vessiot ring of the order two equation 02(y) = - A ( z ) A ( q z ) - i y is equal to P V . The equation r = _ A ( z ) A ( q z ) - i y is seen as a difference equation with parameter q2 instead of q. The ring P V seen as a difference ring with respect t.o r contains a non zero solution of the equation r = _A(z)A(qz)-ty. But P V is not tile Picard-Vessiot ring P V ' of the equation since P V still has the non trivial 02-invariant ideal ( e ( - 1 ) - 1). It can be seen that this ideal is maxinlal amoug the set. of 0"-invariant ideals. Hence P V ' can be identiffed with P l ' / ( e ( - 1 ) - 1). The condition of the criterion is equivalent to P ~ " ~ C(z)[Y, y - l ] , with 02(Y) = - A ( z ) A ( q z ) - l Y and no relations for Y. Using P V ' = P V / ( e ( - 1 ) - 1), one finds that the criterion is also equivalent to G=D~. II In tile second case, tile module C ( t ) ~ M decomposes as a direct sum of two conjugated difference modules C(t)el and C(t)e2 with C-actions given by ~(~i) = (a(, + azt)el and r = (ao - a z t ) e 2 , with ao,al E C(z) and ai # O.

174

CHAPTER

12, q - D I F F E R E N C E

EQUATIONS

The module M is identified with C(z)(el + e2) + C ( z ) t ( e l - c2). The second exterior power A2M has basis tel A e~ over C(z) and is trivial. It follows that ql/2(ao + a l t ) ( a o - alt) = r for some f E C(z). The term ao + a l t may be changed into (ao + a l t ) O ( g ) g -1 with g E C ( t ) ' . Using this one can normalize the choice of ao + alt such that a 2 - za 2 = q-112. The module M is generated over C(z) by the elements el + e2 an t(el - e2). This leads to a matrix equation for M with respect to this basis of the form

al

ql/2ao

y(z).

The matrix equation above, with a0, al C C(z), a0 :~ 0 and ql/2(a~ - za~) = 1, can be seen as the standard form of a q-difference equation over C(z) with group Doo and with a Picard-Vessiot ring containing e(e~ir). In the third case there is a similar standard form with ql/2 replaced by

_ql/2.

Remarks In [27] algorithms for order two q-difference equations are given. In particular the Galois groups of the q-hypergeometric difference equations are determined. There is a theory of q-difference equations for q a root of unity. This theory is developed in [27]. The results are rather different from the case 0 < Iql < 1 studied in this chapter.

Bibliography [1] Abramov,S., Paule, P., Petkovgek, M., q-Hypergeometric Solutions of qDifference Equations, manuscript, 1995. [2] Abramov, S., Bronstein, M., Petko%ek, M., On Polynomial Solutions of Linear Operator Equations, manuscript, 1995. [3] Atiyah, M. and MacDonald, I.G., I n t r o d u c t i o n to C o n m m t a t i v e Algebra, Addison-Wesley, Reading, Mass., 1969. [4] Balser, W., Jurkat, W.B. and Lutz, D.A.,A General Theory of Invariants for Meromorphic Differential Equations; Part [, Formal Invariants, Funk. Ekvacioj, bf 22 (1979), 197-221 [5] Batchelder, P.M., An introduction to Linear Difference Equations Dover Pub, New York, 1967

[6] Benzaghou, B., Alg~bres de Hadamard, Bull. Soc. Math. France, 98 (1970), 209-252. [7] Benzaghou,B. and Bezivin, J.-P., Propridtds algdbriques de suites diffdrenticllement finies, Bull. Soc. Math. France, 120 (1992), 327-346. [8] Bialynicki-Birula, A., On Galois theory of fields with operators, Amer. J. Math. 84, (1962), 89-109. [9] Birkhoff, G.D., General theory of linear difference equations, Trans. Amer. Math Soc. 12, (1911), 243-284. [10] Birkhoff, G.D., The generalized Riemann problem for linear differential equations and the allied problem for linear diffeTvnce and q-difference equations, Proc. Nat. Acad. Sci. 49, (1913), 521-568. [11] Birkhoff, G.D., Theorem concerning the singular points of ordinary linear differential equations, Proc. Nat. Acad. Sci. 1, (1915), 578-581. [12] Birkhoff, G.D., Note on linear difference and differential equations, Proc. Nat. Acad. Sci. 27, (1941), 65-67.

176

BIBLIOGRAPHY

[13] Birkhoff, G.D., Note on a canonical form for the linear q-difference system, Proc. Nat. Aead. Sci. 27. (1941), 218-222. [14] Birkhoff, G.D. and Tritzinsky, W.J., Analgtic theorg of singular difference equations, Acta Math., 60, (1933), 1-89. [15] Bourbaki, N., Alg~bre, Chap. 8, "Modules et Anneaux Semi-Simple", Hermann, Paris, 1958. [16] Braaksma, B.L.J. and Faber, B.F., Multisummabilitg for some classes of difference equations, Preprint University of Groningen. May 1995. [17] Chase, S.U., Harrison, D.K., Rosenberg, A.. Galois theorg and cohomologg of commutative rings, Memoirs of the AMS, 52, American Mathematical Society, Providence, 1965. [18] Coddington, E. and bevinson, N., T h e o r y of O r d i n a r y Differential Equations, McGraw-Hill, New-York, 1955. [19] Cohen, R., Difference Algebra, Tracts in Mathematics, Number 17, Interscience Press, New York, New York, 1965. [20] Deligne, P. and Milne, J.S., Tannakian categories, Lecture Notes in Mathematics, 900, 101-228. [21] Duval, A., Lemmes de Hensel et factorisations formelle pour les opdrateurs aux diffdrences, Funk. Ekv., 26 (1983), 349-368. [22] Etingof, P.I.,Galois groups and connection matrices of q-difference equations, Electronic Research Announcements of the A.M.S., Volume 1, Issue

1, (1995). [23] Fahim, A., Extensions Caloisiennes d'alg~bres diff~rentielles, C.R. Acad. Sci. Paris, t. 314, S~rie I (1992), 1-4. [24] Foster, O., Lectures on R i e m a n n Surfaces, Graduate Texts in Mathematics, 81, Springer-Verlag, Berlin, Heidelberg, 1981. [25] Franke, C.H., Picard- Vessiot theorg of linear homogeneous difference equations, Trans. Am. Math. Soc., 108 (1963), 491-515. [26] Hendriks, P.A., Art algorithm for determining the Galois group of second order linear difference equations, to appear in the J. of Symb. Comp.. [27] Hendriks, P.A., Algebraic aspects of linear differential and difference equations, thesis University of Groningen, november 1996. [28] Hendriks, P.A., van der Put, M., Galois action on solutions of differential equations, J. Symb. Comp., 14 (1995), 559-576. [29] Humphreys, J. E., Linear Algebraic Groups, Second Edition, SpringerVerlag, New York, 1981.

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[30] Immink, G.K., Asymptoties of Analytic D~fference Equatwns, Lecture Notes in Math. 1085, Springer Verlag, Berlin, Heidelberg, New-York, Tokyo, 1984 [31] Immink, G.K., On meromoTphic equivalence of linear dzfferenee operators, Ann. Inst. Fourier, 40 (1990), 683-699. [32] hnmink, G.K., Reduction to canonical forms and the 5'tokes phenomenon in the theory of linear diffe're~ce equations, Siam J. Math. Analysis, 22 (1991), 238-259.

[33] Kaplansky, I., An I n t r o d u c t i o n to Differential Algebra, Second Edition, Hermann, Paris, 1976. [34] Kovacic, J., The inverse problem in the Galois theory of differential fields, Ann. of Math., 89 (1969), 1151-1164. [35] Larson, R.G. and Taft, E.J., The algebraic structure of linearly recursivc sequences under Hadamard product, Isreal J. Math., 72 (1990), 118-132. [36] Levelt, AH.M., Differential Galois theory and tensor products, Indag. Math., New Series 1(4) (1990), 439-450. [37] Loday-Richaud, M., Stokes phenomenon, m.ultisunrmability and differential Galois groups, Ann. Inst. Fourier, 44 (1994), 849-906. [38] Magid, A., Finite generation of class groups of rings of invariants, Proc. Amer. Math.Soc., 60 (1976), 4,5-48. [39] Magid, A., Lectures on Differential Galois Theory, University Lecture Series, Vol. 7, American Mathelnatical Society, (1994). [40] Martinet, J and Ramis, J.-P., Elementary acceleration and rnultisummabil,ty, Ann. de I'I.H.P. 54 (1991), 331-401. [41] Mitschi, C., Singer, M.F., Connected linear algebraic groups as Galois groups, J. of Algebra, 184(1996), 333-361.

[42] NSrlund N.E., Lemons sur les s~ries &Interpolation, Gauthiers Villars et Cie, Paris 1926. [43] Petkovgek, M., Finding closed form sotution,s of difference equations by symbolic methods, thesis, Department of" Computer Science, Carnegie Mellon University, 1990. [44] Petkov~ek, M., Hypergeometric solutions of linear recurrences with polynomial coej~cients, J. Symb. Comp., 14 (1!)92), 243-2(54. [45] Petkovgek, M., A generalization of Gosper's algorithm, Discrete Mathematics, 134 (1994), 125-131. [46] Petkovgek, M., Will, H., Zeilberger, D., A = B , A.K. Peters, Wellsely, Massachusets, 1996.

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[47] Praagman, C., The formal classification of linear difference operators, Proc. Kon. Ned. Ac. Wet. Ser A, 86 (1983), 249-261. [48] Praagman, C., Meromorphic linear difference equations, Thesis, University of Groningen, 1985. [49] van der Put, M., Galoistheorie van differentiaalvergelijkingen unpublished lecture notes (in Dutch), Mathematisch Instituut, University of Groningen, The Netherlands, 1984. [50] van der Put, M., Differential equations in characteristic p, Compositio Mathematica 97 (1995), 227-251. [51] van der Put, M., Singular Complex Differential Equations: An Introduction, Nieuw Archief voor Wiskunde, vierde serie, deel 13 No. 3 (1995), 451-470. [52] Ramis, J.P., About the growth of entire functions solutions of linear algebraic q-difference equations Annales de Fac. des Sciences de Toulouse, S~rie 6, Vol. 1, no 1 (1992), 53-94. [53] Ramis, J.P., About the inverse problem in differential Galois theory: The differential Abhyankar conjecture, in "The Stoke Phenomenon and Hilbert's 16th Problem", World Scientific Publishers, 1996 (editors B.L.J.Braaksma, G.K. Immink, M. van der Put) [54] Renault, G., Alg~bre non c o m m u t a t i v e , Gautier-Villars, Paris, 1975. [55] Reutenauer, C., Sur les gl~ments inversibles de l'alg~bre de Hadamard des sdries rationelles, Bull. Soc. Math. France, 110 (1982), 225-232. [56] Rosenlicht, M., Toroidal algebraic groups, Proc. Amer. Math. Soc., 12, (1961), 984-988. [57] Serre, J.P., Cohomologie Galoisienne, Lecture Notes in Mathematics, Springer-Verlag, New York, 1964. [58] Serre, J.P., Corps Locaux, Hermann, Paris, 1962. [59] Singer, M.F., Algebraic relations among solutions of linear differential equations, Trans. Am. Math. Soc., 295(2) (1986) 753-763. [60] Tretkoff, C., and Tretkoff, M., Solution of the inverse problem of differential Galois theory in the classical case, Amer. J. Math., 101 (1979), 1327-1332. [61] Varadarajan, V.S., Meromorphic differential equations, Expo.Math. 9, (1991), 97-188. [62] Whittaker, E.T., Watson, G.N., A Course of M o d e r n Analysis, Fourth Edition, Cambridge University Press, 1978. [63] Zariski, O. and Samuel, P, C o m m u t a t i v e Algebra, Vol. 1, D. Nostrand, Princeton, NJ, 1958.

Index k-sum, 124 k-summable, 124 p-curvature, 56 q-difference module, 149 formal Galois group, 66 additive problem, 136 asymptotic expansion, 78 asymptotic lift, 133 band, 134 canonical forms, 75 cocycle, 141 cohomology set, 141 connection map, 88, 163 connection matrix, 88 constants, 4 difference ideal, 4 difference module, 24 difference operators, 23 difference ring, 4 exponential torus, 66 fibre functor, 24 formal monodromy, 66 formal Puiseux series, 73 fundamental matrix, 5 Galois correspondence, 21 Galois group, 10 Gamma function, 130 Heisenberg group, 151 holomorphic vector bundle, 158 interlacing, 46 left domain, 78 left flat, 138 lower connection map, 83 lower strip, 131 meromorphic vector bundle, 157 mild, 71, 95

Morita's equivalence, 55 multisum, 96, 125 multisummability, 124, 128 multisummation, 99, 107 Picard-Vessiot field, 22 Picard-Vessiot ring, 5, 7 Puiseux series, 60 quadrant, 129 regular, 71, 156 regular difference equations, 86 regular singular, 71, 152, 154, 156 Riccati equation, 33 right domain, 78 semi-regular of type n, 77, 166 sequences, 45 singular direction, 128 tame difference module, 105 Tannakian group, 87 torsor, 11 total Picard-Vessiot ring, 16 trivial cocycle, 141 unipotent, 60 universal Picard-Vessiot ring, 27, 46, 63, 150 upper connection map, 83, 100 upper strip, 130 very mild, 71 wild, 71

Notations ~,4 CR, 4

c(=), 4 c({~-~}), 4 c((~-~)), 4 P, 4, 72 ,5,,5

e(g)t~it, 97 Iright, 97 lteyt, 97 K, [)7 I<..... 97 ~H+, 98 7/_, 98

f~, 2(5

7~Mild, c.)~ ~.M ild, 98 s 98 H+Mild, 98 ~ _ Mild, 98

g, 60

~o(M), 98

"7, 66 7v, 66 T, 66 k ~ , 69 k ~ , 73 G.~ild, 73 p,~. 77 Ri, b0 Le, 80 Up, 81 Lo, 81 .~o(M), 81 Wright(~4), 82 ~l
Wright(M), 98 wl
D i f f ( k , 0), 24 Vectc, 24 Repr(G), 25

Un~ 8'2

,5'M, upper, 83 SM, lower, 83 A, 84 A ~ 84

d, 86 5'~v/, 87 C,~, 88 C,], 90 "/~, 96 s 96 # ~ , 97

~mild,l, 97 e(g)right, 97

5':,1, w,t,~,', 100 ,q'M. tot,,e,., 100 ~- VeryMild, lO1 g.VeryMild, lO1 1/~(5'), 106

.A4....... 108 A~

117

GM, Io,...al, 120 O(v,k, R), 129 koc, 130 Eq(5'), 141 A'ut(,c;) ~ 141 77'~0, 149 R~,o, I50 G, 150 Tots. 151

a~(M), 1.52 wo(M), 159 w . ( M ) , 159

'lh'iples, 159 M , , 162 /vl r 162, 163 G,,~ - T r i p l e s , 165

T.(M), ~66 H - T r i p l e s , 167