Topics in Galois Theory (Research Notes in Mathematics, Volume 1) (Research Notes in Mathematics)

Topics in Galois Theory Jean-Pierre Serre College de France Paris, France

Notes written by Henri Damon Mathematics Department Princeton University Princeton, New Jersey

ta �

Jones and Bartlett Publishers Boston

London

Contents Foreword

ix

Notation

xi xiii

Introduction

1

Examples in low degree

1. 1 1.2 1.3

The groups Z/2Z, Z/3Z, and S3' The group 04 •

3

•

•

•

•

•

•

•

•

•

exponent 2) 3,4) 6

6

Nilpotent and solvable groups as Galois groups over Q

2.1 2.2

A theorem of Schols�Reich&rdt. The Frattini subgroup of a. finite

.

. . . .

group

.

. . .. . . . . . .

3.2.1 Extension of scalars 3.2.2 Intersections with linear subvarieties Irreducibility theorem and thin sets . . . . Hilb ert ' s irreducibility theorem ..... . Hil bert property and weak approxima.tion Proofs of prop. 3.5.1 and 3.5.2 ......

3.3 3.4 3.5 3.6

.

.

•

.

.

.

.

Galois extensions of Q(T): first examp1es

4.1 The property GalT . . . . . . . . . 4.2 Abelian groups .. . . . . . . . . . 4.3 Example: the quaternion group q. 4.4 Symmetric groups . . . . . . . . . . 4.5 The alternating group At. 4.6 Finding good specializations of T . .

.

v

.

•

.

.. . . . . . . .

.

9

9 16 19

HUb ert's irred udbility theorem The Hilbert property .......... . Propert ies of thin sets ......... .

3.1 3.2

.

4

•

Application of tori to abelian

Galois groups of

2

•

1 1 2

.

19 21 21 22 23 25 27 30 35

35 36 38 39 43 44

Contents

vi

5

6

Galois extensions of Q(T) given by torsion on elliptic curves

5.1 5.2 5.3 5.4 5.5

An a.uxiliary construction Proof of Shihts theorem. A complement.

Further results

on PSL2(F,.)

SL2(F'l)

and

as Galois

gr oups .

Galois extensions of C(T)

6.1 6.2 6.3 6.4

The GAGA principle

.

.

.

.

.

Coverings of Riemann surfaces

From C to Q .

Appendix:

.

.

.

.

.

.

.

.

. . .

.

universal ramified coverings of Riem&nn

with signature 7

. .

.. . . . . .

. . . .

.

.

.

.

.

. .

.

.

. ..

.

.

.

.

.

.

surfaces .

Counting solutions of equa.tions in finite groups

of & family of conjugacy classes .. .. . . The symmetric group Sft The alterna.ting group As The groups PSL2(Fp) The group SL2(Fa) .. . The J anko group J1 The Hall-Janko group J2

Rigidity

7.4.1 7.4.2 7.4.3 7.4.4

7.4.5 7.4.6 7.4.7

.

.

•

•

The main theorem

M

First variant .

8.4.2 8.4.3 8.4.4

.

The symmetric group 8ft The alternating group The group

As

.

PSL2(Fp) ...

The GalT property for the smallest si mple groups

Local properties .

8.4.1

.

Second variant

Examples ......

8.3.1 8.3.2 8.3.3 8.3.4 8.4:

81

.

Two variants .....

8.2.1 8.2.2 8.3

.

.

Preliminaries

.

.

.

.

.

.

.

.

.

.

.

_

.

.

.

.

.

.

.

A problem on good reduction

The

real case

.

.

.

.

.

.

.

..

The p-adic case: a theorem of Harbater

60

78 78

8 Construction of Galois extensions of Q(T) by the rigidity method

8.1 8.2

.

77

•

The Fischer-Griess Monster

.

65 67 70 72 72 73 74 75

.

Examples of rigidity

.

55 55 57 57

65

Rigidity and rationality on finite groups 7.1 Rationality .. .. . . . . . . . . . . .

7.2 7.3 7.4

.

47

47 48 49 52 53

Statement of Shih's theorem .

.

81 83 83 84 85 85 86 87 88 88 88 89 90

93

vii

ContentB

9

form Tr(x2) and its applications Preliminaries ............ 9.1.1 Galois cohomology (mod 2) 9.1.2 Quadratic forma. . 9. L3 Cohomology of 8ft . 9�2 The quadratic form Tr (Z2) 9.3 Application to extensions with Galois group A,.

95

The

9.1

95 95 95 97 98

.

.

.

.

.

•

.

.

.

.

.

.

.

.

•

.

.

.

.

.

.

.

•

.

.

.

.

.

.

.

.

.

.

.

10 Appendix: the large sieve inequality 10.1 Statement of the theorem .

•

.

.

.

.

99 103

10.2 A lemma on finite groups. . . . . . . . 10.3 The Davenport.. Halberatam theorem 10.4 Combining the information . . .

.

103 105 105 107

Bibliography

109

Index

117

Foreword These notes are based on "Topics in Galois Theoryt It

Serre

a.

course given by

J-P.

at Harvard University in the Fall semester of 1988 and written down by

H. Darmon. The coutse focused on the inverse problem of Galois theory: the construction of field extensions having & given finite group G as Galois group, typically over Q but also over fields such as Q(T). Chapter

1

discusses examples for certain groups G of small order.

The

method of Scholz and Reichardt) which works over Q when G is a. p-group

of odd order, is given in chapter 2. Chapter

3

is devoted to the Hilbert irre­

ducibility theorem and its connection with weak approximation and the large sieve inequality. Chapters 4: a.nd

5

describe methods for showing tha.t G is

the Galois group of a regular extension of Q(T) ( one then says that G has property GalT)' Elementary constructions (c.g. when G is a symmctric or alternating group) are given in chapter 4:t while the method of Shih, which works for G = PSL1(P) in some cases, is outlined in chapter

5.

Chapter

6

describes the GAGA principle and the relation bctween the topological and

algebraic fundamental groups of complex curves. Chapters 7 and 8 are devoted to the ra.tionality and rigidity criterions and their application to proving the

property GalT for certain groups (notably, many of the sporadic simple groups,

including the Fischer-Griess Monster). The relation between the Hasse-Witt invariant of the quadratic form Tr (:z:2) and certain embedding problems is the

topic of c hapter 9, and

an

a.pplication to showing that

is given. An appendix (chapter

used in cha.pter

3.

10)

An

haa property GalT

gives a proof of the large sieve inequality

The reader should be warned that most proofs only give the main ideas; details have been left out. Moreover, a number of relevant topics ha.ve

been

omitted, for lack of time (and understanding), namely: a.) Thc theory of generic extensions, d. [Sal].

b) Shafa.revich's theorem On the existence of extensions of Q with

solva.ble Galois group,

d.

a

given

[ILK).

c) The Hurwitz schemes which parametrize extensions with a given Galois group and

a.

given ramification structure,

d.

[Frl], [Fr21, [Ma.31.

d) The computation of explicit equations for extensions with Galois group

ix

Foreword

x

d. [LMJ, [Ma3]J [Ma,4J, [MIl], ... published) on extensions of Q(T) with group 6·�, 6· A." and SL2(F.,). PSL2 (F1)t SL2(F.), MUJ

• •

"'

e) Mestre's results (not yet

We

wish to thank

Larry Washingt on for his helpful comments on an earlier

version of these notes. Paris, August

Galois

1991 H. Darmon

J-P. Serre

Notation If V is an algebraic variety over the field K. and L is an extension of K, we denote by V(L) the set of L-points of V and by l'JL the L-variety obtained from V by base change from K to L. All the varieties are supposed reduced and quasi·projective. A'" is the affine n·space; A'" ( L) = Lft. Pn is the projective n-spa.cej Pn(L) = (L(ft+l) - {O})/L·; the group of a.utomorphisms of Pta is PGLta = GLn/Gm• If X is a finite set, IXI denotes the cardinality of X.

xi

Introduction The question of whether all finite groups can occur as Galois groups of an extension of the rational s (known as the inverse problem of Galois theory) is still unsolved, in spite of substantial progress in recent years. In the 1 9 30 ' s , Emmy Noether proposed the f ollowing stra tegy to attack the inverse problem [Noel: by embedding G in the

S�h

permuta.tion group on n. Q(X). Let E be Galois extension of E with

the

one defines a G·action on the field Q,X1, fixed field under this action. Then Q(X) is a

letters,

• • •

,Xft)

=

Galois group G. In g eometri c terms, the

extension Q(X) of E corresponds to the projection of varieties: 'K : A" --+ A"/G, where Aft is a.ffine n-space over Q. Let P be a. Q-rational point of Aft /0 for which W' is unramified, and lift it to Q E An(Q). The conjugates of Q under the action of Ga.l(Q/Q) are the sQ where s E Hq C G, and Hq is the decomposition group at Q. If Hq G, then Q genera.tes a field extension of Q with Galois group G. A variety is said to be rational over Q 'or Q- rationBl) if it is bira.tiona.lly =

isomorphic over

Q

to the

affine space An

function field is isomorphic to Q(T 1,

• • •

for some

n.,

or equivalently,

[Hi]) If An IG is Q-Tfltio'R4l, then th�Te above such tha.t Hq = O.

Theorem 1 (Hilbert,

ma.ny points P, Q

as

This follows from

Hilbert's

if its

, Tft}, where the T" are indetermina.tes.

irreducibility theorem, d.

Gre

infinitely

§3.4.

Let 0 = Sft, acting on Q(Xh .. . , X,,). The field E of Sn-invariants Tn), where Ti is the ith symmetric polynomial, and Q(X1J"" Xn) has Galois group S" over E: it is the splitting field of the polynomial 2 X"' - T1X"-1 + T2X,,- + ... + (-1)ftT". Hilbert'! irreducibility theorem says tha.t the Ti can be specialized to infinitely many values ta E Q (or even ti E Z) BUch that the equation xn t1X"-1 + t,X"-' + ... + (-1)"� = 0

Example:

is Q(T1,

• • •

)

�

has Galois group Sft OVer Q. In fact, "most" ti with t, E Z, 1 � t, � NJ only O(Nft-\ log N) {Coh], [Se9]. xiii

work: of ma.y

the N" n-tuples (t,)

fail to give 8ft1 d. [Gal,

Introduction

xiv

In addition to the symmetric groups, the method works for the alternating

groups An wi th

n

5 5, cf. [Mae] ( For

n � 6,

it is not known whether the

field of An-invariants is rational.) Somewhat surprisingly, there are groups for which the method fails (i.e. An/G is not Q.rational): •

Swan [Swl] has shown that the field of G-invariants is no t rational when

G

is a cyclic group of order 47. The obstruction is related to the automorphism

group of G which is a cyclic group of order Q((23) does not have class number •

In

1

(since

46 = 2 X 23, and h( -23) 3) .

to the fact that

=

(Le] H. Lenstra gives a general criteri on for the field of G·invariants to be when G is an abelian group: in particular, he shows that this criterion

rational

is not satisfied when G is cyclic of order

8.

(The a.bove counter-examples are over Q. Counter�exa.mples over C (involving

a non-abelian group G) are given by the following result of Saltman [Sa2]: if

there is a non-zero subgroups

H

a

E

H'(G, Q/Z)

ResS} (a) = 0 for all abelian An /G is not C - rati onal . It is

such that

generated by two elements, then

G satisfying

not hard to construct groups

the hypothesis of Saltman's theo­

rem: for example, one may take a suitable extension of abelian groups of type

(P,,,, ,p).)

It is easy to map

see

(e.g., using the normal basis theorem) that the coveri n g 'K =

An

---t

An/G

is generic (or versal) in the sense that every extension of Q (or of any field of characteristic zero) with Galois group G can be obtained by taking the 'l"-fibre

a rational point of An/G over which 'Jl" is unramified. Hence, if AnIG is Q­ rational, then the set of all G-extensions of Q can be described by a syst em of

of

n rational

parameters. Such

a

parametrization implies the following property

of extensions with Galois group G [Sa1]� 2 A.!8ume API /G is Q-rational. Let {Pi} be a finite set of primes, L. eztensions of Q", with Go.lois group G. Then there i8 an eztension L of Q with Gal(L/Q) = G such that L ® QJfj = Li.

Theorem

Remark:

There is a more general statement, where the L, are allowed to

be Galois algebras, a.nd Q is replaced by independent absolute values.

Proor

a field

(sketch): Each Li is parametrized by

parameter

(X)

endowed with finitely many

c�y»

E An(Q) which is sufficiently close to

Q",-topology gives

an

E A "(Q�)'

A uglobal"

ea.ch of the (X(i)) i n

the

extension of Q with group G having the desired local

behaviour (Krasner's lemma ) .

QED.

xv

Introduction

The cyclic group of order 8 does not satisfy the property of tho 2. Indeed, if 2! L Q2 is the unique unr&mified extension of Q:a of degree 8, there is no cyclic extension L of degree 8 over Q such tha.t L2 � L ® Q:a (an easy exercise on characten,

see

[WaD.

perhaps extend Hilbert's theorem to a. more general class of va.­ riet ie s. There is an interesting suggestion of Ekedahl and Colliot.TheIene in this direction [Ek], (CT] (lee S3.5). One could

Since An/G is not always Q- r ati on al, one has to settle for less:

Queztion: If G it G finite group, CAn it be reaJueG 4f tI Gtllou group of some TegUlar ezten.non F oj Q(T) 1 (Recall that '"F is regular" means that FnQ= Q .) Remarkz:

1. H F is a function field of a variety V defined over Q, then F is regular if and only if V is a.bsolutely irreducible. The regularity assumption i. included to rule out uninteresting examples such as the extCIll ion EeT) of Q(T) where E is a Galois extension of Q. 2. If such an F exi.ts, then there are infinitely many linearly di sj oint extensions of Q with Galois group G.

The existence of regular extensions of Q(T) wi th Galois group G is known when G is: • Abelian; • One of the 26 sporadic simple grou p s (with the possible exception of the Mathieu group M:a3); ePSL2(Fp ) , where at least one of is -1 (Shihl1, [Shih2]; • An, or Sn, d. [Hi); • An, d. N. Vila [Vi1 and J-F. Mestre [Me21i • G2(Fp) [Th2], where G2 is the a.utomorphism group of t he octonions, and the list is not exhaustivc.

, (�), (�), (�)

The

method for finding F proceeds as follows:

1. Construction (by analytic and topological methods) of an of C(T) with Galois group G. descent from C to Q. This il the s a.ti sfies a so-called rigidity criterion.

2. A

The outline of the

1.

Elementa.ry

course will

examples,

hardest part, and requires that G

be:

and

the Scholz-Reichardt theorem.

2. Hilbert's irreducibility theorem and

exten sion Fe

applications.

Introduction

3.

4.

The "rigidity method" used to obtain extensions Galois groups.

of Q(T) with given

The quadratic form z ...... Tr (z2), and its applications to embedding problems, e.g., construction of extensions with Galois group An.

Chapter 1 Examples in low degree 1.1

The groups

Z/2Z, Z/3Z,

and

S3

• G = Z/2Z: all quadratic exten sions can be ob tai ned by taking s qu are roots: the map PI --+ PI given by X ..... X2 is generic (in characteristic ditTerent from 2 - for a charaderistic·free equation, one should use X2 - TX + 1 = 0

instead). •

G

=

Z13Z:

A "generic equation" for G is:

3,

with discriminant A = (T2 - 3T + 9)2. (In characteristic the Artin-Schreier equation y3 � Y = -lIT by putting Y = group G acts on Pl by

where

u is a

this reduces to The

l/(X + 1).)

1 X u =l�X'

generator of G .

2 T=X+uX+uX=

.K3-3X+l X2_X

is G-invariant and gives a map Y = Pt --+ PIIG. To check: gmericity, observe that a.nyextension L/ K with cyclic Galois group of order defines a homomorphism � : GK � G � Aut Y which ca.n be viewed as a l-cocycle with values in Aut Y. The extensi on L/K is liven by a rational point on

3

PI/ G if and only if the twist ot Y by this coc:yde has a rational point not

invariant by u. This i.s a general property of Galois twists . But this twist has a rational point over a cubic extension of K, and every curve of genus 0 which

has a point over an odd-degree extension is a projective line, and hen ce has at least one rational point distinct from the ones Jixed by

1

Chapter 1. Examples in

2 •

G

low degree

:;:. S3: The ma.p

gives

a.

projection PI

---+

PI/S3 :;:. PI

which is generic, although the reasomng for 03 cannot be applied, as the order of S3 is even. But 53 can be lifted from PGL2 to GL2, and the vanishing of Hl(GKt GL2) can be used to show that P1

---+

PI! 53

is generic. Exercise: Using the above construction (o r a direct argument), show that every separable cubic extension of K is given by an equation of the form x3 + TX + T::::: 0,

1.2

with T f:. 0,-21/4.

The group 04

Let K4/ K be Galois and cyclic of degree 4, and suppose that Char K 1: 2. The extensi on K" is obtained from a unique tower of quadratic extensions:

J

where K2 = K(.y'E), and K" = K2( IJ, + by'E). Conversely, let K, K( 'V"E) be a quadratic extension of K, where f E Kis not a square. If II, b E K i1.D.d K4 = K2( IJ, + by'E), then K4 may not be Galois over K (its Galois closure could have Galois group isomorphic to D", the dihedral group of order 8). =

Theorem 1.2.1

for some c E K·.

The field K"

.j

is

cyclic of degree 4 if find only if 0.2 - £b"J

=

Ec?

Proof: Let G be a group, £ a non-trivial homomorphism from G to Z/2Z, and X a homomorphism from H Ker E to Z/2Z. Let Hx denote the kernel =

of )(.

Lemma 1.2.2

(a) Hx

is

The following

are

equi1)tdent:

normal in G, and GJ Hx

is

cyclic of order 4.

1.2.

The group 04

(b)

Corix

map

Hl(H)

= E,

3

where

(We ab brevi at e

--+

i it the

Cor

constriction

mllp.

Hl(G� Z/2Z) = Hom (G, Z/2Z) to Hl(G). Hl(G) cm b e defined b y

The corestriction

(Corix}(g) = x(Verig); wh · ere

The proof that ( a) =>

check that the

Now, assume

(b).

is immediate: replacing

(b)

transfer 04

Select

s

a by G/HXt it suffices to by s � 82•

O2 is onto: but this map is given E G - H. The transfer is given by:

--+

Veri( h) = II· shs-1 m�d (H, H). Hence fo r

all II E H:

�

x(Ver h) But if

h

=

==

0

(mod

E

normal in

Hx, then X(h) = O. It follows that X(shs-1) G. Now, applying the hypothesis to B shows X(S2)

so

X(h) + X(shs-1) = E(h)

=

i

Cor x ( s)

s2 :F 1

=

(mod H)(). It follows that le tes the proof of lemma 1.2.2.

com p

Now, let G

=

GK

:::::

E(S)

=:

af Hx

lemma. Via the

where c E

Remark:

K-,

2,

that

H)( is

--.

4,

and

this

define homo�

Hl(GK) with

HI (GK) is equal to the

E�t

This completes the proof of tho

In charac terist ic

K2 a.nd K4

identification of

i

=

10

is cyclic of order

Ke / K-2, the corestriction map Cor : HI (GK:a) norm, and the criterion Cor x = E becomes: N(II + ov'E)

0,

(mod 2).

1

Gal(K /K). The extensions

morphism s C! and X as in the

=

2).

1.2.1.

Art in- Schreier theory gives an iso morphism

where pz = z2 + z, and the c:orestriction map corresponds to the trace. Hence the analogue of tho 1.2.1 in characteristic 2 is:

BI(GK)

=:!

K/pK,

Theorem 1.2.3 Su.ppos e Char K = 2, a.nd let K2 = K (z), K4 K2 (y), where pz = e, py :;;: II + oz . Then K4 it GGlGiI over K Gnd cyclic of degree 4: if lind =

only ifTr ( II + bz)( = b) i3 of the form � + :&2

Observe

that the

variables

E,

4, z of

+

%,

with z

E

K,

tho 1.2,3 parametrize O,,-extensions of

K.

In particular, it is possible in characteristic 2 to embed any quad ratic extension in a cyc lic extension of de gree 4. This is a special case of a general result: the embedding problem for p-groups always has a solu tion in charac teri sti c I' (aa

Chapter 1. Examples in low degree

4 can be seen

from the triviality of H2(G, P) when G is the absolute Galois of characteristic p and P is an abelian p-group with G-actioD.

group of a field

See for example [Sel}.)

The situation is different in characteristic -:I 2: the criterion 0.2 b2€ that E must be a sum of 2 squares in K: jf b2 + c2 #- 0, then: �

implies

e=

(

o.b

b2+c2

) ( 2

+

o.c

b2+c2

)2

::;: fC2

.

v=r E K, and any element of K can be expressed as a sum of 2 Conversely, if «: is the sum of two squares, t = l2 + iJ2, t hen setting

Otherwise squares.

a

solves the equation 0.2

-

=

b2€

�2 + ",2, b =�, =

c2e_ Hence we have

Theorem 1.2.4 A quadratic eztension eztension oj degree

c =

4 if and only if E

K(-v"f)

p.,

shown:

ca.n

be embedded in K.

'" " Bum of two "quare.! in

II

cyclic

tho 1.2.4: the quadratic extension K'J can be 4 if and only if t he homo· Z/2Z given by K2 factors through a homomorphism

Here is an alternate proof of embedded in

a

cyclic extension K4 of degree

e : GK -+ Z/4Z. Thi s suggests that one apply Galois cohomology

morphism

GK

-+

quence:

o

--+

Z/2Z

--+

Z/4Z

-+

Z/2Z

--+

to

the se-

0,

obtaining:

f E Hl(GKJ Z/2Z) to BI(GK, Z/4Z) is given H'l(GKI Z/2Z) = Br:z (K), where Br (K) denotes the 2-torsion in 2 the Brauer group of K. It is well-known that the connecting homomorphism 6 : HI --+ H2 also known as the Bockstein map, is given by 6z = :z: :z: (cup-product). This can be proved by computing on the "universal exam­ K(Z/2Z,1) which is the classifying space for Z/2Z. Th e cup ple" P ao(R) product can be computed by the formula:

The obstruction to lifting

by 6£ E

.

I

=

.

a J3 where

=

(a,{3),

Hl(G,Z/2Z) is identified with K-/K·2 and (a,.B) denotes the class of

the quaternion algebra given by

1.2. The

But

(e, -f )

only if

5

group C. = 0 (in additive notation), so ( ft

(-1, e) = 0, i.e., e

Similarly, one

E)

could ask when the extension

extenlion of degree

8.

=

(-1, f).

il a sum of two squares in

The

obstruction is

One can prove (e,g., by uling

Hence,

K.

6f is 0 if and

K... can be embedded in a cyclic an element of Br2(K),

again given by

[Sea}):

ob.ttM£ction to embedding the cyclic utensioft K4 in eI cyclic =ten.noft 0/ degree 8 it gil1en by the clus of (2, e ) + (-1, eI) in Br2(K)J if eI# 01 elM by the clas" of(2, e) if eI = O.

Theorem 1.2.5 The

Hen ce, when G -# 0, the OSgembedding problem is possible if quaternion algebra (2, e ) is isomorphic to (-1, G ) .

and only if

the

1.2.1. Let K2 and K.. be as before, with + hvlE). Let :z: = K(yE), K... :: K:I( + b..fi. and y = G - bv'E. If Ga.l(K... /K) = 04, then we may choose a generator u of Gal(K4/ K) taking z to 11. and hence 11 to -Z. Setting There i. also a direct proof of tho

K2

Ja

=

Ja

c= we have

(lC =

y( -z)/( -v'i) = c, so

zy/..fi,

c

E

K·, Also

and one obtainl the lame criterion as before. equation (J,2 - b2e = t?e, one verifies that. K... group 0....

Remarks:

1.

The minimal polynomial for z over

x'"

J

K

Conversely, if

at

bt c, e

satisfy the

is a cyclic: extension with Galois

is

+ AXI + B = 0,

A = 2", B = ,,2 - d:l. The condition for a general polynomial of this form to have Galois group C... is that A:I - 4B il not a square and that

where

2. The C,,·extensionl are parametrised by the solutions tion

(J,2

_

d2

=

( E, (J" 1.1 u) of the equa­

E1&:I,

with u � 0 and e not a. square. This represents a rational variety: one can solve for e in terms of ,,' I., and u. Hence the class of C...·extensioDl of Q

6

Chapter

1.

Examples in low d e gree

satisfies the conclusion of tho 2: there are C,,-extensions of Q with arbit r a rily prescribed local behaviour at finitely many placesj recall that this is not true for the cyclic group of order 8. Exercises:

1. The group C4

generator" of C.

t s faithfully on PI via the map C4 - PGL2 which sends a

ac

to (!l �).

The conesponding map PI

-->

4 + 6z2 + 1)/(z(z2 - 1». This gives rise to the equation: 4 Z - TZ3 + 6Z2 + TZ + 1 = 0

by z ...... (Z

PI/C.

i.

gi""D

with Galois group C4 over Q(T). If i E K, show that this equation is generic: in fact, it is equivalent to the Kummer equation. 1.2 If i � K I show that there does not exist any one�dimensiona1 generic family fo r 1.1

C4-extensions. l.3 H a C4-exte nsion is de scrib ed

it

as

before by parameters

comes from the equation above if and only

if -1, a)

(

=

0

E',

a, b a.nd o r a = O.

c,

show that

2. Assume K contains a primitive 2"-th root of unity z. Let L K( 2�) be a cyclic extension of K of degree 2". Show that the obstruction to the embedding of L in a cyclic extension of degree 2"+1 is (a, z) in Br2(K). =

1.3

Application of tori to abelian Galois groups of exponent 2,3,4,6

A K-t oru 8 is an algebraic gro up over K which. becomes isomorphic to a product of multiplicative groups Gm X X Gm over the algebraic closure K of K. If this isomorphism is defined over K, then the tor us is said to be split. Let T be a K· to rus and denote by X(T) its character group, • • .

X(T)

=

Homg ( T, Gm).

It is well known that X(T) i s a free Z-module of rank n = dimT endowed wi t h the natural a ctio n of GK. The functor T ..... X(T) defines an anti-equivalence between the category of finite dimensional tori over K and t he category of free Z-modul� of finite rank with GK action. A split K-to ru s is clearly a K-rational variety; the same holds for tori which split over a quadratic ext ensi o n K' of K. This fol lows from the classification of tori which split over a quadratic extension (whose proof we shall omit - see [CR}): Lemma 1.3.1 A /ree Z-module of finite rank 'With an action of

direct sum of indecomposable modules 0/ the form:

Z/2Z

is a

1.3. Application of tori to abelian G alois

groups of exponent

7

2,3,4,6

1. Z 'With trivial Action.. B. Z with the non-trivicU ACtion. 3. Z x Z with the "regular representation" of Z/2Z which interchan.gea the

two fa.c;tOf's.

The cofTesponding

tori

Are:

1. GlIl !. A "twisted form " of Gm, wh.ich correapcm4s to element.! of norm 1 in (9) below. 9. Th.e algebf'tl.ic gf'Oup RK'/KGm obtained from Gra/K, by "restriction of scAlars" to K ( c/. §9.S.1). It

is not difficult to show that the three cases give rise to K-rational varieties,

a.nd the result follows.

A

HG

=

is a finite group, the group

K[G]

g of invertible elements of the group algebra

an algebraic group

defines

over K.

g�IIG� where the product

is taken

In

ovcr

char&eteristic 0, we

K,

over all irreducible representations of

denote the dimensions of these representations.

In particular, if G i.

Z[G},

where

G

=

wmmutative, then

Homg-(G, Gm).

by the values of the characters of

groups:

and the covering

G. H G

group

m

ap

1

g

--+ --t

G

--+

giG

g splits

and the 71&

There is

an

g --+ gIG

exact

�

4 or 6, then

group

over the field generated sequence

of algebraic

1,

is generic for extensions of

is of exponent 2.3,

G

9 is a torus with character

Therefore,

G.

have

g

K

with Galoi s

splits over a. quadr ati c

Q� Q( �)t or Q(i). By the previous g - and hence " fortiori gIG - is Q-ra.tional. So the abelian groups of exponent 2.3,4 or 6 yield to Noether's method (but not those of exponent 8).

extension, since the characters values lie in

result .

Exercise:

Show tha.t

all tori decomposed by a cyclic extension of

degree 4 are

rational varieties, by making a list of indecomposable integer representations of the

cyclic

[Vo2].

group of order 4 (there are nine of thete,

of degrees 1,1,2,2,3,3,4,4,4). See

Chapter

2

Nilpotent and solvable groups as Galois gron ps over Q A theorem of Scholz-Reichardt

2.1

Our goal will be to Rei chard t [Re):

Theorem

Q.

2.1.1

prove

the fo ll owing theorem which is due to Scholz a.nd

Every I-group, I :/: 21

CAn

be re41ized

(Equivalently, every finite nilpotent group of

over Q.)

odd

AS

A

order

G4iois group over is

a.

Galois group

Remarks! 1.

This is a special case of a. theorem of Sha.farevich: every solvable group can be u a Galois group over Q. [The proofs of tha.t theorem given in [Shall a.nd [Is1 are known to cont ain a. mista.ke rela.tive to the prime 2 (see [Sha.3]). In the notes append.ed to his Collected Papers, p.752, Shafa.revich sketches a. method to correct this. See also [ILKl, ch. 5.J 2. The proof yields somewhat more tha.n the statement of the theorem. For example, IN, then the extension of Q wit h Galois group G can be chosen to be ramified if IGI a.t a.t most N primes . It also follows from the proof that any separable pro-l.group of finite exponent is a. Galois group over Q. 3. The proof does not work for I = 2. It would be interesting to see if there is a wa.y of a.d a.pting it to this I;:ue. 4. It is not known whether there is a regular Galois extension of Q(T) with Galois group G for an arbitrary I-group G. realized

=

An ,...group can be built up from a series of central extensions by groups of orde r I. The natura.! approach to the problem of reali zi ng an I· group G as a Galois group over Q is to construct a tower of ext e nsion s of degree I which ultimately give the desired G·e.xt ension . When carried out naively, this ap p roa.ch does not workt because the embeddi n g proble m cannot always 9

Cha.pter 2. Nilpotent

10

and

solvable groups

be solved. The idea. of Scholz and Reicha.rdt is to introduce more stringent conditions on the extensions which are made at ea.ch stage, ensuring that the embedding problem has a. positive answer. Let K/Q be an extension with Galois group G, where G is a.n l-�oup. Choose N � 1 such that IN is a multiple of the exponent of G, i.e., s' 1 for all s E G. The property introduced by Scholz is the following: =

Definition 2.1.2 The eztension L/Q is said to hAve property (SN) if every prime p 'Which is ra.mified in L/Q satisfies: 1. p ;; 1 (mod IN). §. If tJ is II plAce of L di1liding P, the inertia group Iv at u is equal to the decomposition group Dv. Condition 2 is equivalent to saying that the local extension Lu/Qp is totally ramified, or tha.t its residue field is Fp' Now, let 1

--+

C,

�

G

--+

G

--+

1

be an exa.d sequence of l-groups with Ot ce nt ral . cyclic of order I. The "em­ bedding problem" for G is to find a Galois extension L of K containing L, with isomorphisms Gal(L/ L) � 0, and Gal(i./ K) � G such that the dia.gra.m ---+ -----4

C,

II Gal( L/ L )

G II Gal(£/K)

--+ ---4

--+

G

----+

Gal(L/K)

II

1

is commuta.tive. The Scholz-Reichardt theorem is a consequence of the following (a.pplied indudi vely):

Theorem 2.1.3 Let L/Q be Galois with GAlois group GJ And assume that L h48 property (SN). Assume further that IN is a multiple of tAe e:tpDnent of G. Then the embedding problem for L lind G halJ a solution £, 'Which stJ.tisjies (SN) And is ramified at at most one mon: prime thA'II L. (Furthermore, one can require tha.t this prime be taken from any set of prime numbers of density one.) The proof of tho 2.1.3 will be divided into two parts: firat, for split extensions, then for non-split ones.

First part: the case t;

�

G

X

0,

Let (P1, . . . ,p"') be the prime numbers ramified in L. Select a. prime number with the following properties:

q

1.

q ::::

1 (mod IN),

2.1. A 2. q

theorem

of Scholz-Reichardt

splits completely in the

3. Every prime Pi,

ext ensio n

11

L/Q,

(I S i � m) is an l-th power in FIf'

Taken toget her, these conditions mean that the prime q splits completely in the field L( lV'f,-1P\, ... J�)' The following well-k.nown lemma guarantees the existence of such a. q: LeJIiIna 2.1.4 If E /Q is

II finite eztension of Q, then there aTe in.finitely many primes which split completely in E. 1710 fact, every set of density anI! contains such II prime.

Proof: The second statement in the lemma is a consequence of Chebotarev's density theorem; the first part C&D be proved by a direct argument, without invoking Chebotarev. Assume E is Galois, and let f be a minimal polynomial with integral coefficients of a pri mi t i ve element of E. Suppose there are only finitely many prim es Pi whi ch split completely in E or are ramified. Then fez) is of the form ±p�' .. 'Pkll, for z E Z. When:c is between 1 and X, the number of distin ct values taken by f( z) is at leut .; X. But the number of values of f(:z:) which can be written in the form ±p;"l ... pr' is bounded by a power of log X. This yields a contradiction. H aving ch ose n a momorphism

q

which satisfies the l

:

(Z/qzt

ns above, fix a surjective ho�

conditi o --+ C,.

(Such a � exists because q == 1 (mod I).) We view ). as a Galois character. This defines a Cr-extension MA of Q which is ramified only a.t q, and is linea.rly disjoin t from L. The compo si t um LM), therefore has Galois group G = G xC,. Let us check that LM>. sa.tisfies pro p erty (SN)' By our choice of q, we have q ;:; 1 (mod IN). It r em ai n s to show that Iv Dv at all ramified primes. If p is ramified in L/Qt it splits completely in MA, and hence Dv = I" for all primes vip. The only prime ramified in M>. is q, and q split s completely in L by ass umptio n . Hence, for all primes 11 which are ramified in LM>., we have Dv = Iv a.s desired. =

Second part: the case where G is a non-split extension The pr o of will be carried out in three stages: (i) Existence of an

(ii) Modifying L

ex

so

tensi on i, giving a soluti on to the embedding problem.

tha.t it is ramified at the same pl aces

as

L.

(iii) Modifying L further so that it has property (SN), with at most one additional ramified p rime.

Chapter 2. Nilpotent

12

and

soh-a.ble

groups

(i) Solvability of the embedding problem The field extension L determines a surjective homomorphism; : GQ -+ G. The problem is to lift tP to a homo morphism ¢ : GQ -+ G. (Such a. � is automatically surjective because of our assumption that G does not split.) Let e E H2(G, Cr) be the class of the extension G,and let

tP· : H2(G,O,)

-----+

H2(GQ,e,)

be the homomorphism defined by,p_ The existence of the lifting � is equivalent to the vanishing of tP·(e) in H2(GQ, e,). As usual in Galois cohomology, we write H2( GQ, - ) as H2(Q, -) , and similarly for other fields. The following well·known lemma reduces the statement tP·e = 0 to a purely local question:

Lemma 2.1.5 Th.e restriction mAp W(Q, e,)

--+

:i II H (Q�, 0,) p

is injective.

(A similar result holds for any number fiel d.)

of Proof. Let K = Q(I'I). Since [K : Q] is prime to I, the map H2(Q, e,) -+ H2(K, e,) is injective. Hence, it is enough to prove the lemma with Q replaced by K. In that case, H2(K, e,) is isomorphic to Brl(K), the I-torsion of the Brauer group of K. The l emma then follows from the Brauer­ Hasse-Noether theorem: an element of Br(K} which is 0 locally is O. (Note thatJ since I 'I- 2, the archimedean places can be ignored.) By the a.bove lemma, it suffices to show tha.t ,p*e = 0 locally at all primes. In other words, we must lift t he map tPP : GQ. --+ D� C G to ¢� : GQ. -----+ G. There are two easel:

Sketch

1. pis unramified in L, Le., tPP is trivial on :he inertia group Ip of GQp' Then cPP factors through the quotient GQp/Ip = Z. But one can always lift a map Z -+ G to a map Z ---+ G: just lift the generator of Z.

2. pis ramified in L. By cons truction , 11 == 1 (mod IN), hence p ':f I and Lv/Qp is tamely ramified ( as in 2.1.2, v denotes a place of Labove p); since its Galois group Dv is equal to its inertia group Iv, it is cyclic. The homomorphism QQp -----+ Dv c G factors through the map GQp ---+ Gal(E /Qp), where E is the maximal abelian tame extension of Q� with exponent dividing IN. The ex­ tension E can be described explicitly: it is composed of the uni que unramified extension of Q� of degree IN, (obtained by taking t he fraction field of the ring of Witt vectors over FpiN) and the totally ramified extension Qp( Iyrp) (which

2.1.

is

a.

A theorem of Scholz-Reichardt Kummer extension since p ==

13 1 ( mo d IN) ) .

It follows that

Gal(E/ Qp)

is

a.belian group of type (IN, IN)j it is projective in the category of a.belian groups of exponent dividing IN . The inverse image of Dv in G belongs to that category ( a central extension of a cyclic group is abelian). This shows that a.n

the local lifting is possible.

(ii) Modifying the extension L so t h at it becomes unramified outside ram(L/Q) of primes ramified in L/Q

the set

Lemma 2 . 1 .6

For every prime p, let Ep be

Q,

continuous homomorphism from

Gal( Qp/ Qp) to a finite abelian group O . Suppose that almost IIll £p are un­ ra.mified. Then there is 4 unique f: : Ga1( Q/Q) --.. O J such thtJ.t for all PI the mllps

f:

lind £p IIgree on the inertia groups

1p .

( The decomposition and inertia groups DpI Ip are only defined up to conj ugacy inside GQ . We shall implicitly assume throughout that a fixed place tI has been chosen above each p, so that Dp and

Proof of lemma:

Ip are

well-defined subgroups of GQ . )

By local class field theory, the Ep

Q; �

ca.n

be canonically identified

O. The restrictions of Ep to Z; are trivial on a closed + ,-.. Zp, where n" is the cDnductor of Ep. Since almost all 'np are zero, there is a homomorphism f: : (Z/MZ)· --+ 0, with M = n p"" , and f:(k) = n �(k-l). If we view f: as a Galois character, class field theory shows with maps subgroup 1

that it has the required properties.

(Equivalently, one may use the direct

product decomposition of the ideIe grOup IQ of [Q

=

(I:z; R.i.) X

X

Q,

as:

Q

)

•.

Proposition 2 . 1 . 7 Let 1 -t C -t .. -t CJ -t 1 be a central �zt�nsion of II group • J ana rp he Q, continuous homomorphism from GQ to � which has G lifting .,p ; GQ -.-. i. Let �p : GQ .. -. i be liJtingll of �p = ; I D.. , such that the �p are unra.mified for almost all p. Then there is a lifting � : GQ --+ i such that, for every P, � is equlll to �p on the inertia group at p. Such a lifting is unique. This proposition is

also

P GL", (Tate, see [Se7,

Proof of prop. S. 1 . 7:

useful for relating Galois representations in GL,. and

§6)).)

For every p , there is

a.

Wlique homomorphism

Chapter 2. Nilpotent and solvable groups

14 such that for all which

s

=

..p(s)

Ep(s)�p(s)

a unique E : G Q ---+ C q, = ,pf-1 has the required

E GQ,. . B y the ptevious lemma, there exists

agrees

with Ep on 11"

The homomorphism

property. This proves the existence assertion. The uniqueness is proved simi­

la.rly.

Corollary 2. 1.8 As.!Jumi39 the hypotheses of prop. !. 1 . 7, chosen u3ra.mijied at every prime where ¢ is unra.mijied.

Proof:

Choose local liftings

�" of ¢ which

are

possible since there is no obstruction to lifting

a

unramifie d

tJ

lijti39 of ¢ ca.n be

where

¢

is; this is

homomorphism defined on

Then, apply prop. 2. 1 . 7.

(ii): L

The corollary completes the proof of part is ramified a.t the same places as

L.

Z.

can be modified so tha.t i t

(iii) Modifying i so that i t satisfi es property (SN) We have obtained

an

extension

i which is ramified at the same places as L and G. Let p be in ram(L / Q ) = ram(l/Q).

which solves the extensi�n �roblem for

Ip (resp D", I,,) the decomposition an d inertia. groups for L (resp Ip = Dp C G; this is a cyclic grou p of order lIZ , say. Let I; inverse image of I" in C. We have Ip C D" c I;. If I� is a non-split

Denote by Dp,

L)

at p. We have

be the

extension of I"

( i . e.,

is cyclic of order

the Scholz condition is satisfied

at

p.

lcz+l ) Let

we even have

5

which J� is a split extension of I,,; since ip is cyclic, we

Frobenius element Frob" E

D"I ill

c

Ip

=

be the set of p E

1;1 ip may be

D"

=

I�,

and

r� LIQ) for have I� = ip X C, . The

identified with

an

element

if and only if c" = 1 . If all c,, ' s are equal to 1, i satisfies (SN ) . If not I we need to correct � : GQ -----+ G by a Galois character X : (ZjqZ)· ---+ Cr which satisfies the following proper ties: Cp of e'i the Scholz condition is satisfied at

1.

2.

3.

p

q == 1 (mod IN ). For every p in S, X(p) Cp . The prime q splits completely i n L I Q. =

Conditions 1 , 2, and 3 impose conditions o n the behaviour of

'Vi), Q (�, ¥p, p E 5) ,Nand L respectively. Wri te Q( where F i s cyclic of order I -l and totally ramified at I. Q(

Lemma 2.1A9 The over Q.

fields L, F, and Q (V'f, �, P

E

I�)

q in =

the fields

Q (�) . F ,

S) are linearly disjoint

2.1. A t heorem of Scholz-Reichardt

15

Proof: Since L and F have distinct ramification, L and F are linearly dis­ joint: L · F has Galois group G x C,N-� . The extension Q(�,�, P E S) has Galois group V :;;::: 0, x C, . . . X 0, ( 1 51 times) over q(�i) . The action of Gal( Q(�)/Q) Fi on V by conjugation is the natural action of multiplica­ tion by scalars. The Galois group of Q(�, ..yp, p E S) over Q is a semi-direct product of Fi with V. Since 1 # 2, this group has no quotient of order I: there is no Galois subfield of Q(�' �l P E S) of degree l over q. This implies that L . F and Q( �2 �, p E S) are lin�arly disjoint. QED. =

If S :;;::: {PI ) , " ,P.} , define integers Vi , 2 :5 i :5 k, by C?i = c: . (This is possible if Cl i: 1, which we may assume . ) In order to satisfy conditions 1, 2 and 3, the prime q must have the following behaviour in the extension L · F · Q( � , � , p E S ):

in L · F and q( in q (� , �j in q(� ,

IVt);

.:jP1 /p� ), i

=

2, . . . , k.

By the Chebotarev density theorem and lemma 2. 1.9, such a q exists. One can then define the c�aracter X so that X (Pi) �. T�is completes part (ii): the homomorphism
The proof allows us to generalize the theorem somewhat. Let us ma.ke the following definition:

Definition 2 . 1. 10 If G is a profinite groUPI the following aN: equivalent : 1 . The topology of G is metriz(lble. !. G htU (I cDunteble dense .subset. 3. G C4n be written tU (I denumeroble projective limit

G

=

l�. · ·

-+

G.,.

-+

G.,.-l ...... . . . ) ,

wheN: the G.,. 's 4re finite (a.nd the connecting homomorphisms 4N: surjective).

4 . The set of open .subgroups of G is denumerable. A group G

'Which

sa.tisfie.s these equillalent properties is .said to be separa.ble.

Gal(L/K), these properties are equivalent to [L ; K] � No; if G is a pro- I-group, they are equivalent to dim Hl (G, � No . The proof that the four properties in the definition are equivalent is elemen­ tary.

If G

=

Z/IZ)

Theorem 2 . 1 . 1 1 IfG is 4 separoble p ro- I-group of finite ezponent, theft there is

(I

Galois eztension of q with Galois group G.

16

Chapter

Proof: a finite

If

of ord er

IN

is t he

I-grou p ,

I.

exponent of G, write G

2 . Nilpotent an d solvable

as proj.lim(GR) where

groups

each GR is

the connect ing homomorphi sm b e i ng surj ect ive , with kernel

By tho

2.1 .3,

one

ca.n

G alois extensions LR /Q with

construct inductively an

increasing family of

Galois group G", whi ch have the ( SN) pro pert y;

the union of t he LR's h as G aloi s group G.

Remark: The finiteness condition on the exponent cannot be dropped: for example, Z, X Z, is not a Galois group over A

more

Q.

general. result ha.s been proved by Neukirch [Ne] for pro-solvable

of odd order and finite exponent.

2.2

groups

The Frat tini subgroup of a finite group

Let G be a finite group. Definition

2.2.1 The Frattini su bgrou p 4' of G

mazimal subgroups of G.

�

is the intersection of

the

G. If G1 C G sat i sfi e s G. (Otherwise, choose a maximal subgrou p M such

The Fratt ini subgroup is normal . If G :f:. 1 , then � :f:.

.

G1

=

G,

then G1

:::::

that G1 C M e G. Since . C M, it follows that .Gl C M, which is a cont radi ction . ) In ot her words, a subset of G generates G if and on l y if it gene rat es Glf!: elements of f?, are sometimes referred t o as "non-generators" .

Examples: 1. If G is a simple group, then 4' :::; 1. 2. If G is a p- group , the maximal subgroups are the kernels

of the surjective Cpo Hence f?, is generated by (G, G) and (]J', w here ( OJ 0) denotes the commutator subgroup of G; more precisely� we have

homomorphisms

G

--.

•

= (0, 0) · (]P.

The group 0 / . is the maximal abelian quot ient of G of type (p, p, . . . , p ).

P roposition 2. 2. 2 ( [Hu] , p. 168) Let G he a finite group, • its Frattini subgroup, N (1 normal subgroup of G with 4' c N e G. Assume N/4' is nilpotent. Then N is nilpotent. C o rollary

2.2.3 The group 4' is nilpotent.

This follows by a.pplying prop.

2.2.2

to N

=

•.

Let us p rove prop . 2.2.2. Recall that a finite group is nil pot e nt if and only has only one Sylow p-su bgrou p for every p. Choose a Sylow p-su bgro up P of N, and let Q �P. The ima.ge of Q by the quotient ma.p N -+ N/�

if it

=

2.2. The Frattini subgroup of a finite

group

17

is a Sylow p-subgroup of N/� which is unique b y assu m ptio n. Hence t hi s image is a characteristic subgroup of N/�; i n particular i t is preserved by inner conj ugation by elements of

G, Le., Q is

normal in G . Let

NG(P ) ;::: {g ig E G, gPg-1

=.

P}

be ti.e normalizer of P i n G. If 9

E OT then gPg-l is a Sylow p-subgroup of Q � Applying the Sylow theorem II in Q . there is a q E Q such that

qgPg-l q- l = P. Hence qg E N G( P ). G = N G ( P ), and P is

It follows that G = QNG( P)

normal in

Sylow p-subgroup of N.

Application to

solvable

O. hence in Nj this

=

f)Na( P).

Therefore

implies that P is the only

groups

Proposition 2.2.4 Let G be a finite solvllble group ':1 1 . Then G is isomor­ phic to A quotient oj A group H which is A sem:i.·dirt:ct product U . S, whert: U is a nilpotent normlll sub!JToup oj H, And S is solvAhle with 151 < IGI · Let � be the Frattini subgroup of

Proof:

Gj

since G/� is solvable and #: 1 , it

contains a non-trivial abelian normal subgroup, e. g. , the last non-trivial term

of the descending derived series of G/�. Denote by U its inverse image in G. Since CP c U e G, with U/� abelian, U is nilpotent by prop. 2.2.2. Choose a miLXimal subgroup S of G which does not contain U: this is possible since U #: CP . Since U . 5 ':1 5 and 5 is muimal, G :::: U · S. Hen ce, writing H = U · S (with S acting by conjugation on the normal subgroup U), map H � G.

there is a surjective

The relevance of prop. 2.2.4 to Galois theory lies in the following result

which

s rt s that the embedding problem for split extensions with

as e

kernel has always a solution.

nilpotent

2.2.5 ( [Sha2], [Is]) Let L/ K be lin ezten.non of numher fields with GAlois group 51 let U be (I ftilpotent group with S-action, and let G be the semi-dirt:d product U · S. Theft the embedding prohlem Jor L/ K lind for

Claim

1 -. ha.s

II

U -. G

..... S -+ 1

solution.

Theorem 2.2.6 Cl(lim !. !. S implies the ezistence oj Galois eztensions of Q with given solvo-ble GAlois group.

18

Chapter

2.

Nilpotent and 501va.ble groups

Proof: Let G be a solvable group. We proceed by induction on the order of G. We may asume G # 1. By prop . 2.2.4, write G as a quotient of U . S wi th U nilpotent and S solvable, 151 < 101. The induction hypothesis gives a Galois extension L/Q with Galois group S. By t he claim above, U · S can be realized as a Galois group; hence, so can i t s quotient O.

Let us give a proof of claim 2.2.5 in the elementary case where U is abelian of exponent n. Observe that: 1. If cla.im 2.2.5 is true for an extension L' of L, it is true for L: for I if SI = Gal(L/K), there is a natural quotient map US' -+ US. Hence we may assume I'n C L, where I'n denotes the nth roots of unity. 2. We may also assume u

�

direct swn of copies of Z/nZ [S] ,

because any abelian group of exponent n on which S acts i s a quotient o f such an S-module. Suppose that h is the number of copies in the decomposition of U as a di­ rect sum of S-modules Z/nZ[S). Choose places 111 " , . , Via of K which split completely in L, WI , . , . I w" places of L which extend them; any place of L ex­ tending one of the Vi can be written uniquely &8 OSW�, for some oS E S. Choose elements tPi E L· such that

if oS = 1 and i otherwise.

�,

=

j.

Let M be the field generated over L by the for s E S and j = 1, . . . , h . This i s a Galois extension of K, with Gal (M/ L) � U . Its Galois group over K is a.n extension of S by Uj since U is a free Z/nZ[S}-module, it is known that such an extension splits (see, e.g. , [S e2 , ch. IX] ). Hence Gal(M/ K ) is isomorphic to the semi-direct product of S by U.

Chapter

3

Hilbert 's irreducibility theoreIll 3.1

The Hilb ert property

Fix a ground field K with Char K = 0, and let V be an irreducible algebraic variety over K. (In what follows, algebraic varieties will be tacitly assumed to be integral and quasi-projective.) Denote by V(K) the set of K-ra.tional p oi nt s of V . A subset A of V(K) is said to be of type (01 ) if there is a closed subset W e V, W =f V, with A c W(K), i.e' t if A is not Zariski-dense in V. A subset A of V(K) is said to be of type (C3) if there is an irreducible variety v', with dim V = dim V' , and a generically surjective mor phism 11" : V' --+ V of degree � 2, with A C 1I"(v'(K».

3.1.1

subset A of V( K} is called thin ( �ince " in French) if it is contained in a finite union of sets of type (01 ) or (03), Definition

A

Alternately, a set A is thin if there is a morphism 11" :

havi ng no

ra.tional cross-- section,

Example: If

in K

W -. V

is thin.

Definition

V = Pl , V( K)

3.1.2 (d.

=

with

and such that A C

K U {oo}.

leTS l ] , p.

property if V(K) is not thin.

This is a birational property of Definition

3.1.3

A

dim W � dim V 'If(W( K».

The set of squares (resp. cubes, .

. .

)

189) A variety V over K satufies the Hilbert

V.

field K is Hilbertian if there emu an. i�ducible variety 1, which hu the Hilbert property.

V over K, with dim V �

19

Chapter

20

3.

Hilbert'. irreducibility theorem

(cE. exe rc. 1) that if K is Hilbertian, then the projective line Pl over K has the Hilbert property (hence, our definition is equivalent to the

It is easy to show

standard one, lee e . g.

[L]).

The fields R, Qp are not Hilbertian. A number field is Hilbertian (see

§3.4).

o n irreducible varieties: The variety V is said to be absolutely ir­ reducible if the algebraic closure K' of K in the field K( V) of rational. functions on V is equal to K. Equivalently, V must remain irreducible upon extension

Remark

of scalars to the algebraic closure f< of K. If V is not absolutely irreducible, then V e X ) C W(K), where W is a subvariety of V, W i: V. Indeed, if V is a nonna! variety, then V(K) == 0. For, the residue field of the local ring at

P

E V contains K' , and hence no point of V is K-rational. The genera.l case

follows from this by normalization. In particular, an irreducible variety which has the Hilbert property is absolutely irreducible. Therefore, in our definition of C2-type subsets, we could have asked tha.t V

i

be a.bsolutely irreducible.

� : V' � V be a finite morphism (we also say t hat � is a "covering" even though it can be ramified) . Assume that V and Vi are abso­ lutely irreducible, and let K(V' ) ! K(V) be the corresponding field extension.

Remark: Let I

Let K(V')pl be the Galois closure of K( V' ) over K(V), and let normalis a.tion of V' in K( V')s". The variety W wit h its projection may be called the

Galois closure of V'

gebraically closed in K(V' )Sal, i.e., example, take V = V' = Ph 11'(:)

--+

W need

=

V. Note that K is not

W b e the W --+ V always al­

not be a.bsolutely irreducible.

:3; the Galois closure of V' is

and hence is not absolutely irreducible over X if K does

For

Pl/K(loill)1

not contain 1l3 '

Exercises: 1. Let V be an a:ffi.ne irreducible variety over K, with dim V � 1 . Let WI , . . . , W" be absolutely irreducible coverings of the projective line PI - Show that there exists a morphism I : V --+ PI such that the pullback coverings I·Wi of V are absolutely irreducible. Use this to show that if V has the Hilbert property, then so has Pl ' 2. Let V and T be absolutely irreducible varieties. and A C V(K) a. thin subset. Show that A X T( K) is thin in V X T. M ore generally, let I : W ---t V be a generically surjective morphism whose generic fiber is a.bsolutely irreducible ( Le. the function field extension K(W)jK(V) is regular ) . If B is a subset of W(K) such that I(B) is thin in V(K), show that B is thin in W(K). 3. Let K be a Hilbertian field, and let A be the Bet of elements of K which are sums of two squares. Show that A is not thin. (Use exerc. 2.) 4. Let K be a number field and V an abelian variety over K with dim V � 1. Show that V(K) is thin ( i.e., V does not have the Hilbert property). Hint: llse the Mordell· Well theorem. Problem: H V and V' are irreducible varieties with the Hilbert property, is that V X V' has the same property?

it true

3.2.

Propertiel of thin sets

3.2

Properties of thin sets

3.2.1 Let

21

Extension of scalars

L/ K

be a finite extension,

Extension of scalars to

L

Proposition 3.2.1 If A thin with respect to K .

V

an ablolutely irreducible variety over

yields a. variety over

c

L,

denoted

K.

ViL'

VeL) is thin with respect to L, thm A n V(K) is : (VarL) --to (VarK) from Here are two equivalent definitions

The proof uses the restriction of scalarl functor RLIK

I,.. va.ri eties of RLIK :

to K-varieties,

1.

It il the

for

every

d. [We], [Oe] .

right adj oint to the extension of scalars (VarK) K-variety T an d L·variety W , one has: Mor K(T, RLIKW)

=

�

(VarL),

Le. ,

Mor L(T/L• Wli

In particular, taking T to be a point which is a rational over K , the above formula yields

(RL IK W ) ( K )

2.

=

We L l.

Let E L be the set of embeddings of L in some fixed algebraic closure Kj

E EL, let W" be the variety deduced from the given L-variety W u. Then the product X = It, WIT is a f(-variety. Moreover, one has natural isomorphisms from X to X· for e'1ery s E GK. By a

for each

by extension of scalars via.

K.va.riety from which RLI K W.

Weil's descent theory, these isomorphisms give rile to a

X

comes by extension of scalars; this variety is

type (01 ), then A n V(K) is clearly of type (01), Hence we A C 'If(W(L », that W is absolutely irreducible over L with dim W == dim V, and 'If is a. covering W --+ V with deg 'lf > 1. By restricting suitably V, we ma.y assume that 11' is finite etale. The functor RL/K then gives an etale covering RL1KW � RL/K "iL o Using the diagonal embedding ' A : V � RL/KVL , we obtain an etate covering 'If' : V --+ V, and a Cartesian

If A

is of

may assume that

dia.gram:

The set

' V

An V( K) is contained in 'If' (V ' ( K», and it is easy to check that all the V at least equal to deg 'If . Hence 'If' (V (K) ) and the lame is true for A n V(K).

components of V' ha.ve degree over

is thin,

Corollary

3.2.2 If K is Hilbertillft, then so is

L.

Chapter 3.

22

Hilbert's irreducibility theorem

Suppose L were not. Then A = P 1 ( L) is thin in PI with respect to L. This implies tha.t AnP J ( K) = P l ( K) is thin in PI with respect to Ki contradiction. Remark: The converse to cor. 3.2.2 is not true; see e.g. [Ku1 . 3.2.2

Intersections with linear subvarieties

Let V be the projective space Pn of dimension n, and let A C V( K) be a. thin set. We denote by Grass� the Grassma.nn variety of d-linear subspaces of Pn J where 1 � d � n. 3.2.3 There is 4 non-empty Zari.ski-open subset U such that if W belongs to U(K) J then A n W is thin in W .

Prop o sitio n

C Grass

�

It is enough to prove this when A is either of type � CI ) or of type (C2 ). The first case is easy. In the second, there is a map 11'" : V --+ Pn l with V' absolutely irreduciblet deg ... � 2, and A C ... (V'(K)). By Bertini's theorem (see e.g. , (Jou, ch. I, §6], [Ha, p. 1 79], [De2] , [Z», there exists a non-empty open set U in Grass! such that ... - J (W) is absolutely irreducible for all W E U. Hence, if W E U (K) , then W n A is of type ( C2 ), a.nd hence is thin.

An interesting case occurs when Ii. = 1 . Let '7r : V' ---+ Pn be a generically surjective map of degree > I J and � the hypersurfa.ce of ramification of 1(" . Consi der the set U of lines which intersect c) t ransversal ly a.t smooth points. Then for £ E U, the covering ,... - 1 (£) --+ £ is irreducible: one proves this over C by deforming the line into a generic one, and the general case follows.

Consider the "double plane" V; with equat ion t2 F(:, y), where F is the equation for a smooth quartic curve � in P2• The natural projection

Exam ple:

=

of V; onto P 2 is quadra.tic and ramified along the curve C). A line in P2 which intersects � transversally in 4 points lifts to an irredu ci bl e curve of genus 1 in V;; a line which is tangent at one point and at no other lifts to an irreducible curve of genus zero; fina.lly, if the line is one of the 28 bit angents to � I then its inverse image is two curves of genus zero. In that case, one ca.n take for U the complement of 28 points in P 2 = Grass� . Corollary 3.2.4 If Pn h4S the Hilbert property over K for s om e n � I J then all proiective spaces Pm over K have the Hilbert property. P1 has the Hilbert property over K: if not, P 1 (K) == A i s thin, and A x P I (K ) x . . . x PI(K) is thin in P I x . . . X P I . Th is cannot be the case, since Pn has the Hilbert property (and hence also PI X . . . X PI which is birationally isomorphic to Pn). This impl ies the same for P tn , with m � 1. For, if A = Ptn ( K) is thin i n P tn, then by prop. 3.2.3, there is a line £ such that £, n A L(K ) is thin in £ := Pl . But this contradi cts the fa.ct t h a.t PI (K) has the Hilbert property.

Proof:

=

3.3. Irreducibility theorem and thin sets 3.3

23

Irreducibility theorem and thin sets

V be a Galois covering with Galois group a, where V, W denote varieties, V = WI G, and G acts faithfully on W. Let us say that P E V(K) has property Irr( P) if P is "inert" , i .e., the inverse image of P (in the scheme sense) is one point, i.e. , the affine ring of the fiber is a field Kp ( or, equivalently, GK acts freely and transitively on the X-points of W above Pl. In this case, 'If is etale above P and the field Kp is a Galois extension of

Let

11' :

W

--+

K-irreducible

K w ith Galois group

a.

Proposition 3.3. 1 There is a thin set A C V( K) such that for all P j A, the in-educibility property In( P) is satisfied.

Proof: By removing the ramification locus, we may assume that W � V is etale, i.e. , G acts freely on each fiber. Let E be the set of proper subgroups H of G. We denote by W/H the quotient of W by H, and by 'lfH the natural projection onto V. Let A The set

A

=

U 1I'H(W/H)( K). HE�

is thinl since the degrees of the

'lfH are equal t o [G

:

H]

> 1.

H

P ;. A , then Irr( P) is satisfied! for, lift P to P in W( K), and let H be the subgroup of G consisting of elements 9 E G luch that gF = ,.,F for lome i E

GlC = GaI(K/K).

since the image of P in

H = a. Otherwise, H would belong to E, and W/H is rational over K, the point P would be in A.

Then

If V has the Hilbert property otu!r K, the ezistence 0/ a G­ ootlering W � V (IS above im.plies that there is a Galois e:ctension of K with Gdou group G.

Corollary 3.3.2

A8sume furthermore that W, and hence V. are absolutely irreducible, and that

V has the Hilbert property. Let U8 say tha.t an extension L / K is of twe W if comes from lifting a K-rational point on V to W.

it

Under these (lSsumptions, for etJery finite ezteu.non L of KJ there is a Galois ezteMon E / K of type W with Galois group G which is linea.rly disjoint from L .

Pr op o sition 3.3.3

There e.ut infinitely many linearly disjoint e%ten.sion.s with. Galoif group a, of type W .

Corollary 3.3.4

Cha.pter

24

3. Hilbert ' s irreducibility

theorem

Proof of prop. 9. 9. 9: Extend scalars to L; applying prop. 3.3. 1 there is a thin set AI. C V;L (L) such tha.t for all P ft A, the property Irr( P ) is satisfied over L. Set A :::: AL n V(K). T his is a thin set by prop . 3.2.1 . Choose P E V(K) with P f; A. Then Irr( P) is true over L, and hence a fortiori over K. This P

gi yes the desired extension E.

Exercise: Show tha.t the set of rational points P satisfying property Irr ( P) and giving a fixed G aloi s extension of K is thin (if G � 1). P olynomial interpretation

Let V be as above, let K( V ) be its function field, and let

be an irreducible polynomial over K(V). Let

f

G

c

Sn be the Galois group of

viewed as a group of permutations on the roots of f. (This group can be identified with a subgroup of Sn , up to conj ugacy i n Sn1 i.e., one needs to fix

a labelling of the roots.)

lli( t) E K,

If t

E

V(K)

and

t is no t

a pole of any of the

and one can define the specialization of f at

ft(X) P ro po sitio n

= xn + a l (t)xn-1 + .

3.3. 5 There e2uts a thin set A

then :

1. t is not a pole of any of the 11. f,(X ) is irreducible otler K J 9. the Galois grou, of ft is G.

. .

C

ali,

t:

then

+ an ( t).

V(K) such that, if t � A,

lliJ

V by a dense open subset, we may assume that the a; have no V is smooth and that the discriminant II of f is invertible. The

By replacing

poles, that

au bvariety \If of V X A 1 defined by

( t, 2 ) E \If � ft(z) = is an et ale covering of degree

G. The

•

G

=

53 - Let

over

Its G alois closure W

proposition follows by applying prop.

Examples:

S3

n.

0

3.3. 1

=:

to

YtIN

W.

has G alois group

f(X) = X3 + alX2 + a2 X + a3 be irreducible, with Galois group

K( V) .

The specialization a.t t has Galois group 53

properties are satisfied:

if

the following

1. t E V(K) is not a pole for any of the a., . 2. 6(t) -:f. 0 ( where 6 = a�a� + 1 8al a2a3 - 4a�a3 - 4a� - 27a�

discriminant

3.

of f).

Xl + al(t )X2 + a 2 (t)X + alet) has no root in K.

is the

25

3.4. Hilbert's irreducibility theorem

4. 6.(t) is not a square in K. Conditions 3 and 4 guarantee tha.t G is not contained in either of the maximal subgroup. of Sa, namely 82 and A3• It is clear that the set A of t E V(K) which fail to satisfy these conditions is thin. • G == 54 . The maximal subgroups of G are Sa, A&, and Dol, the dihedral group of order 8. The case of G c S3 or A& can be disposed of by imposing the same conditi on. as in the case G = S3i to handle the case of D.. , one requires the cubic resolvent

( X - (ZlZ2 + :1:3:1:.. » ( X - (Zl:1:3 + Z2:1:4» (X - (2:IZ4 + Z2Z3)) 3 X - 4,X 2 + (4143 - 4(4)X + (44244 - 4� 41 - � )

=

to ha.ve no root in K J where ZlJ

• • •

, Z.. are the roots of the polynomial

• G = Ss . The ma.ximal subgroupi of G are S4t S2 X S3. and As , which give conditions similar to the above, and the Frobenius group F20 of order 20, which is a lemi-direct product OsC4 and can be viewed as the group of affi ne linear transformations of the form z ...... 4Z + b on Z/5Z. If the Galois group of the specialized polynomial is contained in F2Ch then the sextic resolvent, which is the minimal polynomial over K(V) for

( :C I Z2 + Z22:3 + Z32:4 + Z4:1:S + ZSZl ) - (:1:1:1:3 + Z2:t4 n,<j(Zi - z;)

+

Z3:1:S + Z4 2: 1 + ZSZ2)

ha.s a root in K when specialized.

3 .4

Hilb ert 's irreducibility theorem

Theorem 3 . 4 . 1 (Hilbert [Hi)) II K is

a number field, then jor every n1 the 4ffine space A· ( or equivalently the projective space Pta) has the Hilbert prop­ erty over K . (In other words, K is Hilbertian.)

By cor. 3.2.2 and cor. 3.2.4, it is enough to show tha.t the projective line P1 over Q has the Hilbert property. There are several ways of proving this: 1. Hilbert's original method ([Hi]): The proof uses PWleux expansions (cf.

(1)). It showl that if A C A l (Q) is thin, then the number of integers n E Z n A with n < N is D( N' ) for some 6 < 1 , when N -. 00. (But it does not give a. good estimate for 6.)

2. Proof by counting points of height smaller than N on P l (Q}: write e E PI (Q) as (:I:, y) with z , y E Z and :I: , y relatively prime. This can be done in a.

Cha.pter

26

3.

Hilbert 's irreducibility theorem

unique wa.y, up to a choice of sign. The height of ( is defined to be height(!) The number of points in P l ( Q ) � N 2 . On the other hand: Prop osition 3.4.2

height

� N

is

<: NJ

=

wi t h

sup( I :t I, Iyl) . height less than

N

is asymp totically

If A C Pl (Q) is thin, then the number of points of A with when N ---. 00 .

Sketch 0/ proof: We may assume A i s of type (02 ), A c ,,"(X(Q» , where X is is an absolutely irreducible curve and ?I" : X ---4 PI has degree � 2. Let Ratx (N) be the number of points :t in X(Q) with height(?r(:t» :5 H, and let 9 be the genus of X. Case 1: 9 � 2. Then X(Q) is finite by Faltings's theoremj hence Rat x ( N ) is

0(1). One could also invoke an earlier result of Mumford (d. [L] ,[Se9] ) which gives Ratx( N) = O(loS log N ) . Case 2 : 9 = 1: B y the Mordell-Weil theorem, and the Neron- Tate theory of normalized heights, (see [ L] ) , one has Ratx(N) ::::: O« log N)-r/ 2 ) , where 7 is the rank of X(Q). Case 3: g = 0: let Hx I H denote the heights on X and Pl respectively. It is known tha.t Hx ;::.. (H 0 1r )tn, wher e m = deg 1r . H en ce , the number of points on X with H 0 'K most O( N1/m ) .

� N

is at

3. Proof by counting integral points: A variant of the second proof shows that the number of integral points in A with height len than N is O ( N t ), which is an optimal bound. 4. Proof by counting S-unih: let S = {Pi, . . . , PA: } be a. finite non-emp ty set of primes, Es = {±p;"L . . . p:'I1 } , ?'nt E Z. Let 0 E Q, and let 0 + Es be the set of 0 + e, where e E Es . If A is a thin s et in P1 (Q) J then A n (0 + Es ) finite number of o .

Proposition 3.4.3

for all but

a

is

finit e

Recall that, if V i s an affine variety over Q , a subset B of V(Q ) i s called quasi-S-integral H for every regular function f on V, the set fe B ) has bounded denominators in the ring of S-integers, i.e., there is a non-zero integer (J (de­ pending on f and B) such that Bf(b) is an S-in tege r for all b E B . Let now ?r; : Xi --+ PI be a finite number of coverings. and choose a outside the finite sets ram(wj ) of X at wh i ch 7fi is ram ifie d . If Bj C X; (Q) is the set of elements of Xj(Q) with '1rJ(b) E 0 + Es , then Bj is a. quasi-S-integral

3.5.

Hilbert property and �a.k a.pproximation

27

- ':1r;I (OO) - ""'i l (a). Over Q, 11"; 1 (00) U 11";1(£1) has at least 3 elements, since ""'il (a) f. ram(wj). A theorem of Siegel, Mahler &Dd Lang then shows that Bi is finite ([L], [Se9 ) ) . Prop . 3.4.3 follows.

set in the affine curve Xi

Remark: The bound given in proof no. 3 can be. extended to affi ne n-space More precisely, let A be a. thin set in An(Q), and let Int A(N) be the number of integral points (:l: 1 , . . . , :en) E A with 1 :1:1 1 $' N. Then:

An.

Theorem 3.4.4 ( S . D . Cohen) IntA(N)

=

O(N"- \ 10g N ) .

One c an replace the log N term in this inequality b y (log N)", where .., < 1 is a const ant depending on A. The proof is based on the large sieve inequality: one combines tho 3.6.2 with cor. 10. 1 . 2 of the a.ppendix, d. [Coh} and [Se9] .

Let X C Pn be an absolutely irreducible variety of dimension r. As above, denote by Ratx ( N ) the number of points of X(Q) with height :5 N. IT X is linear, deg X = 1 t then Ratx ( N ) ;::. N�+ l . P roblem:

IT

deg X � 2, one (:an

show, using

t ho

Ratx( N)

3.4.4 =

(d.

[Se9J), tha.t

O ( N�+\ log N).

A better result follows from results of Schmidt [Sc:hm]. namely, Ratx Can

thil eltimate b e improved to: Ratx (N)

3.5

=

( N) = O ( N,,+t ).

O ( N1'+C) , for every

t > 01

Hilbert property and weak approximat ion

Let K be a number field, EK the set of places in K (including the archimedean ones). For v e EK , let K. denote the completion of K at v , and let Nv be the cardinality of the residue field of K. in case v is non-archimedean. If V is an absolutely irreducible integral v-ariety o"er K, V(K,,) is

naturally

en dowed

with a Kv-topology which gives it the structure of a. K.-analytic: space (resp. manifold, if V is smooth).

Proposition 3.5.1 If W C V, w 1: V, then W( Kv) 1: V(KII ) for all DU,t finite number of v E EK •

a

Chapter 3. Hilbert's irreducibility theorem

28

Pro p osition 3.5 .2 Let W be absolutely irreducible, of same dimension

W

as

V,

deg 1r > 1 . Let Kf( be the algebraic closure of K in the. ezte.nsion K(W)gal / K ( V ) . If 'U E EK splits completely in K'Jf and Nv is large enough, then 1r(W(K,,» #- V( K,, ) . and

1r :

The proofs

Exam p le:

an d define

�

V

be a generically S"U1iecti'fJe morphism,

of prop. 3.5. 1 and Take K

1r :

W

=

---.

Q, V

3. 5 . 2 will be

gi ven in § 3.6.

= PI, W the curve defined by

y3 = ( x 2 + 3)/(,2:2 + 12) ,

V by 'K ( :Z: . y ) = :z: . The n Kf( = Q ( yC'3) . 1 (mod 3). If p == - l (mod 3 ) , p � 5, then

A prime

p

K.". when p == one checks that 7r : W ( Qp) -+ V(Qp) is an isomorphism of analytic Qp.manifolds . This shows that the con d i t i on ICV splits completely in K/' cannot be omi t te d. splits in

Theorem 3 . 5 . 3 Let A be a thin subse.t of V(K ), and let So b e. a finite subset of �K" then there is a finite set S of places of K satisfying : 0. a) s n so b) The image of A in llues V(K,, ) is not dense. =

Observe first that if the theorem holds for A l and A2, it holds also for Al U A2: for, choose Sl sat i sfyi n g the conclusion of the theorem for Al i then choose 52 satisfying the conclusion So U Sl '

Then, taking S

is not dense: the

=

of the theorem Sl U S21 one has

point ( :& 1 1 :1:2 ) ,

for A2• b ut with So rep laced by

where :Z:i E nve s.

in the closure of A l U A 2 i n ilv e s V(K1J ) ' Hence i t is enough t o prove tho 3.5.3 for

V(K., )

- closure(Ai ), is not

sets of type (el ) and (02),

V, W #- V, A C W ( K ) , then ch oo se S {v} with v large enough , ;;. SOl W ( Kv) #- V(K,, ) (prop. 3.5. 1 ) . Since A. C W( Kv), A i s n o t dense in W ( K., ) .

1.

If

W

so that

C

v

=

Ass ume A C

'K(W(K»), where 1r : W ---t V i s generically surj ective. dim V I W is absolutely irreducible1 and deg 1f � 2. We ma.y also assume t hat 1f is a finite morphism (replace V by a s ui t ab le open subvariety) . 2.

dim W

=

a v ;;. So such tha.t 1r(W(Kv» =1= V(Kv). Since closed subsets into closed sub sets. T his shows th at 1r(W(K,,)) is closed in V(Kv)j hence A is no t dense in V(K,, ) . By

1f

prop. 3.5 .2, there exists

is finite, it transforms

3.5.

Hilbert property and

29

weak approximation

Corollary 3.5.4 Assume

V is

CIi

projective 11arietYt

ancl let A

denote the clo­

SUre of A in the compact space

II V(K.,) .

• t!So

Th.en the interior of A is empty, i. e., A is nowhere dense in the product.

One says that V has the weak appro:rimation property for a fini.te if V( K) is dense in nVES V( Kv ) .

set S of

places

Lemma

3.5 . 5 If V, V' are smooth, birationally equil1C1ilent, then V has the

'Weak approzimation property for S if and only i.f V hu the weak appro i2:ima­ non property for S. (In other words, the weak approximation property is a

birational property for smooth varieties).

It is enough to prove the lemma when Vi = V -W, with W a closed subvariety, W # V. Clearly, if V has the weak approximation p roperty for S, so does V - W . Conversely, if V - W has the weak approximation property for S , one uses smoothness to prove that V(KtI) - W(K,,) is dense in V(K,,). Hence, any point in W(Kv} can be approximated by points in V(K) - W(K}. A s a special. case, any smooth K-rational variety has the weak approximation property for any finite set S of places. Remark: T he smoothness assumption is necessary: for example, consider the Q-rational curve 'Jl

=

(:1:2 - 5)2 ( 2 - :1:2 ) .

Its rational points are not dense in the set of its real points (the points ( - \1'5, 0 ) (../5, 0) are isolated).

a.nd

Definition 3.5.6 A 11ariety V is said to have property (WA) if it satisfies the weal approzimation property with respect to S for

CIilI finite

S e EK .

It is

property (WWA) ( "wed wed approzimation propertyn) if there a finite set So of pla.ces of K such thClit V ho.s the weak approximation

said to have

e=z:iats property with respect to S e EK, for all S with S n So

Examples:

1. A K·rational

=

0.

variety has property (WA).

2 . A K-torus has property (WWA), but not necessarily (WA). More precisely, if it

is split by a. finite Galois extension L/ K, one can take for exceptional the places of K whose decomposition group in Gal( L/ K) is not cyclic. (See, e.g. , [Vol),[CTSl) . ) set

So

Chapter

30

3.

Hilbert's irreducibility theorem

Theorem 3.5 . 7 ([Ek] ,[CT] ) A l1ariety which has the WWA property satisfies the Hubert property. V( K ) would be thi n . B y tho 3.5.3 there would exist Proof: If not , A S disjoint from So such that V(K) would not be dense in TIues V(K,,); this contradicts WWA. =

The followi ng conjecture is due to Colliot-TheHme [CT] i it is closely relate d to the questions discussed in [CTS2] :

Conjecture 3.5.8 A K -uninltional smooth variety has the

property.

WWA

Recall t hat a variety is K-unirational if there exists a generically surj ect ive map Pn --+ V defined over K , for some n - one may always take n equal to dim V.

Theorem 3.5.9 Conjecture 9. 5.8 implies that group over Q .

every

finite group

is

a.

Galoia

Proof" Make G act faithfully on W = Aft for some n, and let V = W/ G. Then V is K-unirational. By conject ure 3.5.8, V ·lnOotb has the WWA p roperty, and hence satisfies the Hilbert property. B y cor. 3 . 3.2, G can. be realized as a Galois group over Q. 3.6

Proofs of prop . 3 . 5 . 1 and 3 . 5 . 2

Let 0 denote the ring of i ntegers of K , an d choose a sch eme .l::': of finite t y pe over 0 having V as its generic fiber; any two choices for V coin ci de outside a finite set of primes, i . e ., they become isomorphic as schemes over Spe cO[ � ] for some non-zero d. If 'l1 is a. non-archimedea.n pla.ce, p. t he corresponding prime of 0, denot e by It(v ) = O/P" the residue field a.t 'lit which is a. finite field with N'I) elements . Let V( It( v)) be the set of ,,(v ). rat ional points of � (or, equivalently, of the fiber of V at 'l1). We shall use the followi ng known result:

Theorem 3.6. 1 (La.ng-Weill If V

is

absolub!ly irreducible over K then J

The ori g inal proof of Lang-Wei I is by re du cti on to the case of curves for which one can use the bound proved by Weil. A different method, which gives a mor e precise error term, is to use DeligIle's estimates for the eigenvalues of the Frob enius endomorphism, together with Bombieri's bounds for the number of zeros and poles of the zeta function, see [Bo] .

3.6.

Prooti of prop.

3.5.1

and

3.5.2

31

Proof of p r o p . 3.5.1 We ma.y assume that V is smooth and W = 0 (by replacing V by V - W). If N'IJ is large enough, tho 3.6. 1 implies that V( It('IJ» is not empty. Choose � E V(It('IJ». Since V is smooth at v (for N'IJ large enough). :t lifts to a point :E E V(Ov), which is contained in V( Ku). Hence V(Kw ) is not empty.

Proof of prop. 3.5.2 Recall that we are given a generically surjective map 11" : W --+ V, with deg 11" > I, dim W = dim V. By replacing W and V by open subsets if nec­ essary, we may assum e that X' is finite etale and V is smooth. We may also choose a scheme of finite typ e j£ for W over some O[�1, such that 'Ir comes from a map (also denoted 'Ir ) If: ...... V. By changing d, we may further assume that 'lr : W --+ J[ is etale and finite and V is smooth (see, e.g. EGA IV 18). For v prime to tI, we have a diagram W(It('IJ»

f-

V(It('IJ»

f-

�

1£(0.) �

V(Ou)

'-+

W(K. )

'-+

V(K. ).

�

The fact that 'Ir : W --+ V is finite implies that the right square is Cartesian, i.e. a point Z E W(K. ) is an O.-point if and only if 11'(%) E V(K. ) has the same property. M oreove r, the reduction map V(O.) --+ V« 'IJ» is surjective, since V is smooth. To show that W(Kv) -:F V(K.) it is thus enough to prove that W(It('IJ» -:F V( It('IJ» . This is

a

consequence of the following more precise result:

Theorem 3.6.2 K�I one has

when

with

c= 1

c' <

Let

m = deg .,.. (m �

2) . ThenJ for

'11

splitting completely in

- �I ' (This implies:

1, for N'IJ large enough, '11 splitting completely in Kw.)

Let 1I::s.J be the Galois closure of j£; its Galois group G injects i nto S", I an d hence IGI S m!. Now, divide the points in }£:(It('IJ)) into two Bets:

Chapter 3. Hilbert's irreducibility theorem

32

where A

.

I At + I B !

H

1.1

a.t v

By tho 3.6. 1 applied to W, we have: =

b solu tely irreducible,

a.

all

the connected components of t he

letting

and hence,

components, we ha.ve

IWsal(�( 1.1»1

We ha.ve this gives

=

=

I;'

e(Nv)di:mV + O« Nv)d.im V - i ). H

m

Hence :

Wral( It ( v» gives

Nv )dhnV- l ).

5

..!.IAI + IB I + O« Nv )dim V - l )

�

( Nv )dim V - (1 -

m

� )IAI + O« m

Nu )di m V- : ) ,

and therefore

Finally, since

(1 -

acts freely,

=

� IAI + O« 'r(A)1 + Ir( B ) 1

Since

(N1.1)dimV + O« NtI )d.im V - � ) .

The same argument applied to the act i on of G on

IT( A )I

�) I�\

�

fiber

e be the number of these

A = WIIII ( It(v))/ H where H = Gal(WralIW). lA '

and B is the

( Nv )dim V + O« Nv)dbn V - t ) .

s plit s completely in K'ff' 1 then are

wralc"c v )),

is the set of points which can be lifted to

set of the remaining p oints

"'�HI

'/r(W(It(u») � (1 -

�

I�I � ,!!

I

we get :

--\ )(NV )diD1V m.

+

O« Nu )dim V - i ) ,

and this completes the proof of tho 3. 6.2, and hence of prop. 3.5.2.

3.6. Proof. of prop_ 3.5 . 1

and

3.5.2

33

A p plication to the distribution or Frobenius elements

Let let

E be

a.n

dliptic curve over

ap, for p prime, be the

Q without complex multiplication over QJ 4p

N� is

where

the

and

"trace of Frobenius" : =

1 + p - Np

nu mber of point s of B. over F�. The following

is

well-known:

3.8.3 1/ / 1: 0 is Any polyn.omial in t'Wo vAriables over Q, then. the set 0/ primes p such thGt J(p, ap) = 0 ha.s de1L!lity O.

Theorem

( The

proof uses the I-aclic representa.tion

p : GQ

---+

GL2(ZI)

a.ttached

to

since det p(Frobp) = p, Tr p( Frob. ) = ap, one is lead to consider the set of :z: E p(GQ) such that J(Tr z , det :z:) = O. Since p(GQ) is known to be open

E:

GL2(Z,), this .et has Haar measure zero; the theorem follows by Chebotarev'. density theorem. )

in

More

generally, 3.8.4 Let A be G thin subset of Z A ha.s d.ensity O.

Theorem

(p, tip )

E

applying

X

Z. The set oj p 's such th4t

is already proved for A of type (01): so assume A is of type (02 ). Let A, be the image of A in ZIIZ x Z/lZ and let 5, be the set of � E GL (Z/IZ) such that (Tr ( . ) . det ( � » bdongs to A, . Qne checks that 2 I S. I :5 I A,I . l2 (1 + 1 11). The theorem

By tho 3.6.2, we ha.ve

J AIl 5

c12 for c <

completely in some fixed extension K of

Is, l ::; This shows tha.t the density

S

of S,

1 Qi

crt { l +

in

and

I sufficiently

large,

splitting

hence

Ill).

GL2(ZIIZ)

c( l + 1/1)(1 - 1/Zt1 ( 1 - 1 /12 r1

is =

c(l - 1 /1r

2

' " , 1m are large enough distinct pri me points of E is n GL,(Z/l.;Z). H each 'i splits completely in K the Chebotarev density theorem, a.pplied to the field of 11 l",- division points of E, shows tha.t the upper density of the 2 set of primes p with (p, lip) E A is S cf'4 n�l (l - 1 /ls)- . Since this can b e m.ade arbitrarily small by taking m large enough, t he theorem follows.

if 1 is large enough. Recall now that, if 11 , numbers, the Ga.lois group of the 11 • . •

• •

- i.-division

Chapter 3. Hilbert'.

34

irreducibility

Remar ks:

theorem

.

1.

The above implies, for example, that the set of the primes p for which E(F, )I - 3 is a square has density O. I 2. One can prove more than density zero in 3.6.4j in fact, one has

1{P l p < N, (p , a.)

E

A}I = 0

This implies that

E

( log �)1+' ) ' 1

P (P,op)EA

<

for

some 6

00.

The proof (unpublished) uses the Selberg UA2 " sieve. 3. There are similar rellults for the Ramanujan 'T-function.

>

O.

Chapter

4

Galois extensions of examples 4. 1

Q (T) :

first

The property GalT

Let E be a. finite Galois extension of Q(T) with group G which is regular, i.e., Q n E = Q. Geometrically, E can be viewed as the function field of a smooth proj ective curve 0 which is absolutely irreducible over Q; the inclusion Q(T) '--+ E corresponds to a (ramified) Galois covering C --+ PI defined over Q with group G.

Conjecture

4. 1 . 1 Every finite group

G

covering.

occurs a.s the Go.lois group of such a

Let us say that G has property GalT if there is a regular G-covering C -t P 1 as above. In that case, there are infinitely many linearly disjoint extensions of Q J with Galois group G (d. tho 3.3.3). Remark: If a regular G-covering exists over Pta , n � 1 , then such a covering also exists over Ph by Bertini's theorem (d. e.g. [Jou] ).

Examples:

The

property GalT is satisfied for:

1. Abelian groups. 2. � and 9ft (Hilbert);

.A. (Vila, Mestre) . 3. Some non-abelian simple groups, such as the sporadic ones (with the possible exception of M23), most PSL2(F,), p prime, and a few others. 4. If G has property GalT, then so does every quotient of G. Proposition

G1

X

4. 1 .2 If

GI , G2

h.ave property

GalT}

then so does their prodtJ.t:t

G2 •

Let Ci t O2 be regula.r coverings of PI with groups Gl t G2 , and let Ih , E2 be their ramification loci.

Proof:

35

Chapter 4. Galois extensions of Q (T)

36

1. If EI n E2 = 0, then the extensions corresponding to Ch C2 are linearly disjoint, because PI i s algebraically simply connected ( see §4.4 below). One can take for covering C the fibered product Cl X PL Ch which has function field Q( CI ) ®Q(T) Q( C2). and hence has 01 x O2 as Galois group. 2. If El n E2 f:. 0, one modifies the covering 01 ----t P I by composing it with an a.utomorphism of PI s o that the new ramification locus is disjoint from E2• One is thus reduced to case 1. Remark : If G 1 and G2 have property GalT, one can show (cf. e.g. [Ma3), p. 229, Zusatz 1) that the wreath product 01 Wr G'l al s o has property GalT.

This

can

be used to give

an

alternate proof of prop. 4. 2.2 below.

Exercises:

1. Show that the profinite group Zp is not the Galois group of any regular extension of Q(T) . (Hence conjecture 4. 1.1 does not extend to profinite groups , not even when they are p-- adic Lie groups .)

2. Let G

b e a. finit e group having property GalT. Show that there exists a. regular Galois extension L of Q(T), with Galois group G, such that: ( a) Every subextension of L distinct from Q(T) has genus � 2. (b) Every Q�rational point P of PI has property Irr(P) with respect to L. (Use a suitable base change PI - Ph combined. with Fattings's theorem.)

4.2

Abelian groups

A torus defined over Q is said to be a "permutation torus" if its character group has a Z-basi s which is st able under the action of Gal(Q /Q) 1 or equiValently, if it can be expressed as a product of tori of the form RKi/QGm , where the K. are finite extensions of Q. A permutation torus is clearly rational over Q. Now J let A be a finite abelian group. The following proposition implies that A has property GalT: Prop osition

seq),

4.2 . 1 There emu a torus

such. that the quotient S'

S' is a Q-rational variety. )

=

SIA is

S

a

over

QJ

and an embedding of A in

permut4tion torus.

(In particular,

The proof uses the functor Y which to a torus associates the Z-dual of its character group. An exact sequence of the form 1 --+ A --+ S --+ S' ---4 1 gives rise to the exact sequence 1 where

----t

Y (S)

-----to

Y (S')

-----to

A

--to

1,

A = Extl (A, Z) = Hom( � , A) = H om( A. , Q/ Z ) ,

4.2. Abelian group s

37

and A denotes as usual the Cartier dual Hom ( A, Gm). Choose K finite Galois over Q such that the action of GQ on A factors through Gal(K/Q), e.g. Q(Pn), where n is the exponent of A. Now express A as a quotient of K a free Z[Gal(K/Q)]Mmodule F, and let S' be a torus such th a.t Y(s') = F; it follows t h at S' is a permutation torus, and there i. an AMisogeny S --t � . =

Proposition 4.2 . 2 Ltd G be a finite grou.p having property G alT 1 and let M be a finite abelian group with. G- action. Th en the semi-direct product G = M · G tUso ha property GalT.

We may assume without loss of generality = e gA, 80 th at

M

IIEG

G = ..(A

X

• • •

that M is an induced G-module,

X A) · G

101 ti me8

is the urreath. product of A and G. By prop. 4.2. 1, there is an isogeny 5 ---+ 5' defined over Q, with S' a p ermut at ion torus and with kernel A C S( Q). By hypothesis, there is a regul ar etale G-covering C

--+

U where U is a Q-rational

variety ( e.g. an open subvariety of PI)' The actions of A on S, and of G on C and on S X x S give rise to a natural a.. action on X = S x . . . x S x C. T his action is free . Let Y = X/a. P rop . 4.2.2 then follows from the following . • •

lemma.:

Lemma 4 . 2 . 3 The variety

Y

=

X/G is Q-rational.

' 5 x Define X' = X/(A x . . . x A) . x s' x C . We have Y = X ' / G. This show s that Y is the fiber spa.ce over U with fiber the torus s' x . . . x s't =

"

which i s asso ci ated t o the princip al G.. bundle C --+ U. We may thus view Y as a torrLS O1ler U. In particular t the generic fiber Yu of Y � U is a torus x s' by a G­ over the function field Q ( U) (this torus is obtained from s' X twisting. using t he Galois extension Q( C)/Q(U» . Since S' is a permutation torus over Q. Yu is a permutation torus over Q ( U). Hence the function field Q ( Y) of Yu is a. pure transcendental ex tension of Q( U ) , which it self is a pure transcendental extension of Q. The lemma follows. (One can also deduce lemma 4.2.3 from lemma 4.3.1 below.) • • •

generated. by an abelian normal subgroup M and subgroup G having property GalT. Show usin g prop. 4.2.2 that H has property

Exercise: Let H be a finite group

a

GalT-

Ch a.pte r 4. Galois extensions of Q(T)

38

4 .. 3

Example: the quat ernion group

We need

first:

Lemma 4.3.1

Let

fiber W . ThefJ.

E is

Y

--+

Q8

X be an etale Galois covering with Galois group

G J end G --+ GL(W) be e lineer representation of G, where W is a finite­ dimensional vector space. Let E be the associated fiber bundle with base X and biratiofJ.allti isomorphic to X x W .

This follows from Hilbert's theorem 90: Hl (K(X). GLft) = 0 , where n is dim W, and K(X) is the function field of X over the ground field (which we assume to b e Q ) .

[Alternate proof: Use descent theory to show that E is a vector bundle over X hence is lcxally triviaL] This lemma. implies that Q(E) = Q (X)( TI I " " Tn), hence that Q(E) and Q(X) are stably isomorphic. (Recall that two extensions kl a.nd le2 of a fiel d Ie are stably isomorphic if there exist integers nI , n:z � 0 such that the extensions k1 (TJ , ' . . , Tn1 ) and �(Tl " ' " T", ) are Ie-isomorphic. ) I

Application:

Let G act linearly on vector spaces WI a.nd Wa , n = dim WI , we have:

W2 being faithful. Letting

4.3.2

Q( WI x W2 ) / G � Q ( W2 / G ) (T1 ,

• • •

(In particular. Q ( W1 x W2 )/G i s

consequence Corollary

the action on

, Tn).

stably isomorphic to Q (Wz /G). ) of the lemma ap plied to E = WI X W:z and X = W,.

4.3.3

If

G

a.cts faithfully on

Q(Wz/G) are sta.bly isomorphic.

T hi s

is

a

WI and W2 , then Q ( W1 / G) and

e Wi b e a d e compos i tion of the regular representation of G as a sum of Q·i rred ucible ones. By the corolla.ry, if one of the Wi is a faithful G-module, then Q(R/G) is stably isomorphic to Q (Wt / G). If Q ( Wi/G) is a rati on al fiel d , then so is Q ( R/G), by 4. 3.2. Let R

=

Qs. Let Q s be group algebra Q (Qa] de com p oses as

Application to the quat ernion group

group of order 8.

The

Q[Qa]

=

the

qua.ternion

Q x Q x Q x Q x H,

where H d enotes the standard field of quaternions (over Q). and Qs a.ct s on H by left multiplication. By t he previous remark , the Q-rationality of Q [Q.]JQs is equival ent to the Q- rationality of H· /Qa) where H· is the mul tipli c ati ve group of H viewe d as a 4-dimensiona.l Q-algebra.ic group . The group Qa has

4.4.

Symmetric groups

a center

{±1}

of order

39

2,

and

D

=

Q s /{±l} ia a. group of type

(2, 2).

On the

other hand, by the map

(N, 4J) : H·

--.

Gill X S 03,

where N is the reduced nonn, and tP : H· ---+ SOa mapa a quaternion % to -l the rotation y � � y� (on the 3-dimen8ional vector apace of quaterniona of trace

0).

Therefore, H·/Q a

and it suffices to show that

Gill

SOa/ D

x

SOa/D,

is a ra.tional variety over

Q. But the

the stabilizer in S03 of a flag in A:I. Hence 503/ D is isomorphic to &n open subvariety of the flag variety of A3 t which is rational over Q . Noether's method therefore applies to Q s ; in particular Q a has property GalT .

group

D ia

=

{JY].

For explicit formulae, s ee

Exercise (L. Schneps) Show that every p-group of order r has property GaiT (use the exer cise at the end of J4.2). 4.4

Symmetric groups

The symmetric group Sft acts on the a.ffi.ne apace A'" with quotient A'" ( "sym­ metric functions theorem" ) . This shows that 9.,. has property GalT. Let give some explicit constructions of polynomials with S.,.

as

UB

Galois group. For

example, consider a polynomial

and put

I(X, T) Theorem 4. 4.1

(Hilbert

(The polynomial

1

of f(X, T) over Q(T) is

a

=

X" + Ct1 X"- 1 + . . . + 4,.

-

T.

[Hi] ) If 1

is G Morse function, then the splittif&!J field regulGr eztension with Galois group S.,. .

ia called a Morse function if:

1. The zeros {jl t . . . , (j.,.-l of the 2 . !C{3i ) #: 1(11; ) for i t= j .)

derivative

t of f are simple.

We will need the following simple fa.cts about the symmetric group Sn :

Lemma 4.4.2 Sft

is g

en.era ted by tratUpositions.

Chapter 4. Galois extensions of Q(T)

40 This is well-known: indeed

S,.

is generated by the tra.nspositions

( 1 2), (2 3 ) , . . . , (n - 1, n ) .

Lemma 4.4.3 Let G be

a transitive subgroup of S,. which contains a transpo­ sition. Then the follo wing are equivalent : 1. G contains an ( n - 1 )-cyde. !. G is doubly tra.nsitil1e. 9. G = S,. .

H G contains an ( n - 1 )-cycle, then the stabilizer of a point is transitive on the complement of the point, hence G i s doubly transitive . IT G is doubly transitive, then G contains all the transpositions in Sn. , hence G = S,. by lemma 4.4.2. That 3 => 1 is obvious.

Lemma 4.4.4 (d. {Hul, p. 171 ) Let G be

a tra.nsitive subgroup of Sn which is genemted by I!'JIdes of prime orders. Then: 1. G is primiti11l!l. !. 1/ G contains G tra.11Sposition, then G = Sn ' 9. If G contains a 3-cyde, then G = An. or Sn. .

Let {y) , . . . ) lk}, wi t h Ie > 1, be a partition of { l , . . . I n} which is stable under G. Our assumptions imply that there is a cycle s of G, of prime order p, such U sP-1Yi is fixed by s, we that Yi #:- sYi. Since no element of Yi U sYi U have: IYi I + I s Yi I + . . . + I sP- 1 Yi I � p, . . •

and hence l Yi I = 1 . This shows that { Yi , . . . , Yi:} is the trivial partition of {I, . . . , n} . Hence G is primitive. To show 2, let 0' be the subgroup of G generated by the transpositions belonging to G. Since Gr f:. 1 , it is transitive ( a non·trivial normal subgroup of a. primitive group is transitive). For {} C { l , . n} , let us denote by Sn (respo An) the symmetric (resp , altem&ting) group on O. Let 0 C { i , . . . I n } be maximal with the property Sn c G', and suppose tha.t n f=. { I , . . . I n}. By the transitivity of G' , there exists (%y) E d with % E O J Y r;. n. H enc e Snu{lI} c d , contradicting the maximality of {}. It follows that {} = {i, . . . t n} and hence G = 0' = Sn proving 2. The proof of 3 is similar 1 t akin g G' this time to be the subgroup of G genera.ted by the 3-cycles belonging to a. The hypothesis implies that a' is non-tri vial, &nd hence is transitive. Choose 0 C { I , . . . , n } maximal with the property An C a'. As before, if Q :f:. {I, . . . , n } , there is a. 3- cy cle ( x yz) which does not sta.bilize O. There are two cases: ' Case 1 : {:Z:, Yt z} n O has two elements, say y and z. Then clearly Anu{a} C O ) contradicting the maximality assumption for O. 0 . ,

I

4.4. Symmetric groups

41

Case 2: {=) lI� Z} n n hu

1 element, say z.

Choose two elements ,/, z' E n

(xyz) and (:ty' z') generate the alternating {=, Y . Z, lI', ZI}. In particular, the cycle (zy' %) is in G; since thi s

distinct from Zj it is easy to see that

group As on

3-cycle meets n in two elements, we are reduced t o cue I , QED. More generally, Jordan has shown that a primitive su.bgroup of Sft which contains a cycle of prime order � Th.

4.4.1

n

-

3 is equal to An or Sn (see [Wi] , p.39).

will be proved in the following more general form:

4.4.5 If K ill Any field of chArGcteristic 01 or of chAmcteristic p not di1liding ft, And f(X) E K[X] is Morse, the" Gal(f(X ) - T) = Sft over K(T).

Theorem

Proof.

We may usume

be viewed

&8

K

to be algebraically closed. The polynomial

a ramified covering of degree

f:

PI

Z

---+

The corresponding field extension is

l-+

n

X=

can

PI

t ::; fez).

K(X)

:J

K(T);

it is separable because

p Jn. Let G C Sft be the Galois group of the Galois closure K(T), i.e. , the Galois group of the equation f(X) - T = O. The ramification points of the covering f are

At

f

of

K(X)

over

00, the ramification is tame, and the inertia group is generated by an

n-cycle. At the fUJi), the hypothesis on I implies that the inerti a. group is tame for p :f:. 2 and wild for p = 2, and that (in both cases) it is generated by a transposition.

Hence the theorem is a CO,nsequence of the following propositionl combined with lemma 4.4.4.

P roposition 4.4.6 Let C

----+ PI be A regulAr Ga.lois covering with group G, tAlJUly ramified At 00. Then G is genemted by the inertiA subgroups of points outside 00, And their conjug4tu.

Prool:

Let H be the normal subgroup generated by the inertia subgroups

outside 00.

Then C/ H is

a

GI H·covering

of PI which is tame at 001 and

unramified outside. The Riemann-Hurwitz formula implies that the genus of

O/H is � l(l - IG/H!)i hence G = H.

Corollary 4.4.7 A) P I is a.lgebmica.lly simply connected. b) In chAmcterUtic 0, the 4ffine line is 4lgebra.iCGlly nmply connected.

Chapter 4. Galois extensions of Q(T)

42

In characteristic p > 0 , the affine line Al is not simply connected (as shown e.g. by the Artin·Schreier equation XP - X T). H G is the Galois

Remark:

=

group of an unramified covering of A 1 , then prop. 4.4.6 implies that G is generated by its Sylow p-subgroups. There is a conjecture by Abhyankar [Abl] that, conversely, every group G ha.ving this property occur• . This is known to be true when G is solvable, and in many other cases (Nori, Abhyankar, Harbater, Raynaud, see e.g. [Ab2] , {SeU]). Example

of an

8n - ext e nsi on

of ramification

type

( n, n - 1, 2)

The above method for constructing polynomials over Q(T) with Galois group Sn gives us polynomials with ramification type (n, 2, . . . , 2). For a different example, consider the polynomial

I(X) so that

I(X, T)

=

=

Th en

xn - xn- 1 ,

xn

- )("- 1 - T. a =

and hence the ramification is given (in char. 0) by

{

Hence

G

=

(n, n - 1, 2).

Remarks:

at 00 : at 0 � at a :

The polynomial I has ramification type

Xp+t - XP - T. This polynomial has Sp+l

as Galois group in characteristic different from p (the proof is similar to the one above) . In characteristic p, one c an show that i t h as Galois group PGL2(Fp ). 2 . One might ask for an explicit polynomial In over Q such that In. has GaloiB group Sn.. Here is an example: l"eX) = xn. - X - 1. Indeed, Selmer [Sell has shown that In is irreducible over Q. Assuming this, let us prove that In has Galois group 8ft • We look at the primes p dividing the discriminant of In, i.e . , those modulo which I.,.. ( X) hu a multiple root. This happens if In(X) and f�(X) = nXn-1 - 1 have a common root mod p. Substituting xn- l = l/n in the equation I(X ) == 0, one gets X ,:::::: n/(l - n). Hence there can be at most one double root mod p for each ramified prime p. This shows that the inertia subgroup at p is either trivial, or is of order two, generated by a transposition. But G Gal(/) is generated by its inertia subgroups, because Q has no non­ trivial unramified extension. By Selmer's result, G is transitive; we have just shown that G is generated by transpositions; hence G = Sn by lemma 4.4.4. =

=

n

cycle of order nj cycle of or?� r n - 1 i a transpositlon.

Sn. by lemma 4.4.3.

1. Consider leX, T)

1

1 - -,

43

At.

4.5. The alternating group

Many more examples can be found in the litera.ture. For instance, the "trunca.ted exponential" Z2

Z3

l + z + 2" + "6 + " ' + z,. n�

(mod 4), and Galois group � otherwis e

haa Galois group S", when n :/: O

(1. Schur).

Exercises: 1.

Let Y,.

1.1.

of P,.- l

be the subvariety

Y,. is

an

94

=

for i =

0, 9&

==

4. 96

=

=

1 +

(n - 2)!

49, 9T

=

48 1,

1 .2. The quotient of Y,. by Sn. ( actin g by

to P1• 1.3.

The Galois covering

given by

1, 2, .

absolutely irreducible smooth curve, 9n.

(e.g. 93, =

e homogeneous equations

defined by th

Y,.

--+

PI

• .

whose

n2 - 5n + 2 4 • . •

show

genus 9n.

is

given by:

.

)

permut ation of coordina.tes) is isomorphic

is the Galois closure

the polynomial I(X) = xn. - x,.- l .

2. In ciaaracteristic: 1 1 ,

, n - 2.

of the

degree

n

covering

that the equation

is an unramified extension of A l whoae Galois closure has for Galois group Mathieu group Ml l . (Hint: reduce mod 1 1 the equa.tion of [Ma.41 , after divi�ing X-variable by

4.5

the

the

111/4.)

The alternating group A n

One exhibits the alterna.ting group An as a Galois group over Q(T) by using following lemma ( " do uble group t rick" ) .

the

Lem m a 4.5 .1 Let G be the Galois group oj G regulAr eztension K/k(T), ram­

ified At most tit three plAces which Are ra.tional over k, and let H be tJ. subgro'Up of G of inde� !. Then the ped field Kl of H is rAtional. (In particular, if

k

=

Q, then

H

has property GalT.)

Beca.use of the conditions on the ramification, the curve corresponding to Kl has genus sero, and has a k-rational point. The lemma follows.

Chapter 4.

44

G aloi s

extensions of

Q(T)

For example, the polynomials with ramification typ e ( n t n - 1 , 2 ) di s cus s ed can be used to cons tru ct An-extensions of Q ( T ) . More precisely, let us change variables, an d p ut in the previous se ction

he X, T)

=

(n - l)xn - nXn-1 +

T.

Then the discrimina.nt of h (with respect to X) is

Up to square factors, we have

�( h) ......

{

(- l)T(n - l)(T - 1) ",-t

(- 1 )2 nT(T - 1)

if n is even;

if n is odd.

Hence the equa.tion D2 == � define s a rational curve. For ex a.mple , even, by replacing T by 1 + ( -1)i {n - 1 )T2 J we get the equation

(n _ 1 )xn - nXn- 1 + 1

+

(-l)T ( n - 1)T2

=

if

n

is

0,

gives rise to a regu l ar Galois extension of Q ( T) with Galois group An . Hilbert's original construction [Hi] was somewhat different. For th e sake of simplicity, we reproduce it here only in the case where n = 2m is even. Choose a polynomial

which

g(X)

=

m-l

nX II (X - 1',) 2 ,

with the Pi distinct and nonftzero. Then, take f(X) so that df/dX g. A ss ume that the fUh) are a.ll di stinct , and distinct from 1(0). Then f h as ramification type (n, 2, 3, 3, . . . , 3). Hence its Galois gr oup is S by lemma n 4.4.4. But then the quadr&tjc subfield fixed by An is only ramified at the two places 00 and 0; hence it is a rational field. =

Exereise:

4. 6

Show that the

condition that the f({3,) are t-

1(0) can be suppressed.

Finding goo d specializat ions of T

Let I be a polynomial over Q( T) with splitting field a. regul ar G·extenlion of Q ( T ) . Although Hilbert's irreducibility theorem guaranteel that for "most" values of t, the speciali zed polynomial I(X, t) will have Galois group G over Q, it does not give a constructive method for finding , say. an infi ni te number of su ch t's.

4.6.

Finding good

specializ&tions of T

45

For p f S, where S denotes a suitable finite set of pri mes , the equation I( X, T) = 0 can be reduced mod p. If p is large enough (see exercise be­ low ) , then all conj ug ac y classes in G occur as Frobenius elements at t for some t E Fp. Letting C1, , CIt be the conjuga.cy classes of G J one can thu s find. distinct primes PI , . . . , PI. and points th . . . J t�, with ti E Flti • such that FrobJPf (f(X, t.» = Ci . Specializing T to any t E Q su ch that t == ti (mod Pi) for all i gives a polynomial f( X ) = f(X, t) whose G alois group over Q int er­ sects each of the conjugacy classes Ci, and hence is equal to G, by the following elementary result� • • •

Lemma 4.6. 1 (Jordan, [J2] ) Let G be G finite group, and H G which meets every conju.gAcy cla,ss oj G. Then H G.

a

subgroup 0/

=

Indeed, if H has a.t most

is a subgroup of G'I then the union of the conjugates of H in G

1 + ( G : H) ( l H I - l )

=

I G I - « G : H) - 1 )

elements. Exercises: 1. Use lemma 4.6.1 to show that every fini te division

algebra is commutative (d. Bourbaki, A.Vm, §111 no.l). 2. Show th&t lemma 4.6.1 can b e reformulated as: "Every transitive subgroup of Sn, n � 2, contains a permutation without fixed point" . (This is how the lemma is stated in Jordan, [J2].) 3. Let 11" : X ----t Y b e a. G-covering of absolutely irreducible projective smooth curves over Fp' Let N be the Dumber of geometric points of X where " is ramified, and let 9 be the genus of X. As su me that 1 + p - 2g';p > N. Show tha.t, for every conjugacy class c: in G, there is a. point t E Y(F,,) over which 'If is unramified, and whOle Frobeni us class in G is c. (Apply WeiPs bound to the curve X twisted by an

element

of c.)

I

5

Chapter

G alois extensions of Q (T ) given by torsion on ellipt ic curves 5.1

Statement of Shih's theorem

Consider an elliptic curve E over Q(T) with i-invariant equal to T, e.g. the curve defined by the equation 2

y + zy - z _

3

-T

36 1728 z

_

-T

_

1 1 72 8 '

(Any other choice of E differs from this one by a quadratic twist only. ) By a.djoining to Q(T ) the coordinates of the n-division points of E, one obtains a Galois extension Kn of Q(T) with Gal(Kn/Q(T» = GL2(Z/nZ). More precisely, the Galois group of C · Kn over C(T) is SL2(Z/nZ ), and the homo� morphism GQ(T) ---+ G L2(Z/nZ ) � (Z/nZ)· is the cyclotomic character. Hence the extension Kn is not regular when n > 2: the algebraic closure of Q in Kn il Q(Pn). So the method does not give regular extensions of Q(T) with Galois group PGL2(F.. ) nor P SL2(Fp). Nevertheless, K.y. Shih was able to obtain the fonowing result [Shihl], [Shih2] :

.

Theorem 5 . 1 . 1 There ezists " regultl.f" eztension of Q( T ) 'With Go.lois P S L2 ( Fp) if = -1. = -1, ; = - 1, o r

(�)

()

(�)

Shih's theorem will b e proved in §5.3.

Remark: It is also known that P SL2(F..) has property Ga.h when this follows from [M12] combined with tho 5. 1 . 1 .

47

(�)

group

=

-

1;

Chapter

48

5.2

5.

Elliptic curves

An auxiliary const ruction

Let E be an elliptic curve defined over a field k of characteristic O. The p­ torsion of E, denoted E [P], is a two.dimensional Fp.vector spacej let us call PE[P] the associated projective line P E[P) � P l (Fp). The actions of the Galois group Gle on E[P] and P E(P] give rise to representations P

and

:

p : G"

Ole

--+

-+

GL(E[PD

PG L( P E [P))

�

�

GL 3 (F,) PGL 3 ( Fp ).

The determinant gives a well-defined homomorphism PGL2(Fp ) --+ F;/F;3 . The group F;/F;2 is of order 2 when p is odd (which we assume from now on ) . Hence the homomorphism Ep det op is a quadratic Galois character, =

fp :

GK

�

{± 1 } .

The Weil pairing gives a canonical identification of J\ 3 ( E[PD with IJp , so that det p ::::; X, where X is th e pth cyclotomic chara.cter giving the action of G, on the pth roots of unity. Therefore Ep is the quadra.tic character associated to k(#), where p. (-l )�p. Let K be a qua.dratic extension of Je, and let tr be the involution in Gal( K/k) . Let N be an integer � 1 and E an elliptic curve defined over K such that E and E' are N-isogenous (i .e. , there is a homomorphism � : E --+ E' with cyclic kernel of order N). Assume for simplicity that E has no complex multiplication, so that there are only two N-isogenies, ¢ and - � , from E to =

E',

If pIN, the maps �, -¢ induce isomorphisms (also denoted ¢, -t/J) from E[P] to E'[P] . By passing to the projective lines , one gets an isomorphism ¢ : P E[P] -+ P E' [P] which is independent of the choice of the N-isogeny E -+ E', and commutes with the action of GK. We define a map PE,N : GIr. --+ PG L (P E[PD as follows: 1. If oS belongs to the subgroup GK of GIr., then PE,N(S) is defined via the natural action of GK on P E[P] .

2. If oS E Gle - GK, then the image of S in OK/It is a. Hence s gives an isomorphism PE[P] --+ PE'[P], and p(s) : PE[P] -+ PE[P] is defined by composing this isomorphism with �- l : P E'[P] --+ P E[P]. describes the action of G, o n the pairs of points ( z , y ) in P E[P] X P E' [P] which correspond to each other under the isogeny �. ) One checks easily that the map (In other words,

p E,N

PE,N

�

Olr.

-+

PGL(PE[PD

5.3. Proof of Shih'. theorem

49

so defined is a homomorphism.

As before, the projective representation p ai N of Gle EBJI

:

Gle

---to

gives

a

character

±1.

Let EK denote the character Gle --to ±1 corresponding to the quadratic exten­ sion K/le. Recall that fp : Gle --to ±l corresponds t o k{H)/k.

Proposition 5 .2.1 We have :

(�) if (�) and (;: ) ;::: if

Corollary 5.2.2 1/ K image of PIl,N

u

=

le( #),

=

1

= -1

-lJ then EIl.N

contained in PSL2 (Fp) .

=

I, i. e.,

tAe

Proof of 5.1. 1 : If s E GK, then PB.N(S) acts on A2 E[P] � ".." by Xes). Hence = f,t.( .. ) . If .J f/. GK, then using the fact that the homomor­ EB,N(S) phism t/I- l : Ecr [pl --+ Efp] induces multiplication by N-1 on =

(�)

( where the group operation on IJ.p is written additively), one finds that

det PE.N(of)

Hence EB,N(.!)

5 .3

=

=

N-1X(of).

(�) Est(s). This completes the proof.

P roof of Shih's theorem

Let KN denote the function field of the modular curve

Q,

Xo(N) of level N

over

KN =: Q[jh j 21! FN(it , j1),

where FN is the (normalized) polynomial relating the i-invariants of N-iso. genous elliptic curves (cf. IF]). The Fricke involution WN acts on KN by interchanging ;l and ; 2 . Let Xo(N}P be the twist of Xo(N) by the quadratic character Ep t using the involution WN• The function field of Xo(N)P can be de­ scribed as follows: let leN be the field of invariants of WN in KN t i.e., the £UDc�

ti on field of Xo(N)+. Then KN and I:N( H) are disjoint quadratic extensions

50 of

Chapter

leN .

The third quadratic extension, Ie, of

t.he function field of Xo (N)"

leN contained

in

5. Elliptic

K

=

curves

KN ( H) is

Now, let E denote an elliptic curve over K wit h j-invariant. jl ' Then Ell' has = -I, j- invari ant j2, and hence E and Ell' are N-isogenous. Assuming cor. 5.2.2 gives a projective representat.ion Gle ---t PSL2(Fp l . It is not hard to see that. this representation is surjective) and gives a regular extension of le.

(�)

Remark: Instead of taking E ove r K one can take it over KN. One thus gets projective representa.tion Gle N ----+ PGL2 (F,,) and prop 5.2. 1 shows that it is surj ective and gives a regular extens ion if C� ) = - 1. Since kN is known to be isomorphic to Q(T} if N belongs to the set P a

.

,

s=

{2 . 3 � 5 , 7. 11, 13, 1 7, 19 , 23, 29, 31, 41 , 471 59, 71},

this shows that PGL2(Fp ) has property GalT provided there exists N E S with = - 1 (a computer search shows tha.t this is true for all p < 5329271).

(�)

We

are interested i n values o f N for

which le

such th at :

Xo(N) has genus O.

1. 2.

Xo(N)P has

Assuming

that

-

a Q rat ional N

is

not a

i s i s omor p hic t o

Q (T),

i.e. ,

point .

square,

condition 1

implies that N = 21 3 , 5 , 6, = 2 , 3 , 7 ( and also for cf. {Shihl] ) . More precisely:

7, 8, 1 0, 12, 13, 18. Condition 2 is satisfied when N N

=

6,

(: )

=

1 and

N

=

1 0,

Proposition 5.3 . 1 For N a re

=

(; )

::;; I ,

2, 3, or 7, the two JUed points of WN in Xo (N)

rational over Q.

(Hence, these points stay rational on Xo ( N )P, which is thus isomorphic to

over Qj this concludes the proof of tho 5.1 . 1 . ) Let uS give t wo proofs of p rop

. 5.3.1.

P1

First proof: Let 0 b e an order of class number 1 in a quadra.tic imaginary field, a.nd Eo an elli ptic curve with endomorphism ring O. Assume that the unique prime ramified in the field is N I and that 0 contains an element '1r with '1r7r � N. This is indeed possible fo r N = 2, 3. 7:

5. 3 . Proof of Shih's theorem

51

N

0

?r

2

Z[iJ

1 +i

Z [v'-2]

v'-2

Z [ 1 +0] Z[v'=3]

3

Z [�]

7

Z [v'=71

vC3 v=a

.;=7 v'=7

By the theory of complex multiplication, t he pair ( Eo , ?r ) then defines a ratio­ po int of Xo (N) which is fixed under WH.

nal

Second

prool: Let 6. be the discriminant modular form ) q ;;::: e2ft· , A( q) = q II(l - qft 24,

and let f be the f

power series

= =

defined by

(6(z)/A(Nz) lf.::r q-l II ( 1 - q" )2m ,

ft� 1

".O(H)

where m =

12

--

N- l

.

One can prove t hat I generates KN over Q, and that fWN = Nm / f. For N = 2, 3, and 7 we have m = 12, 6, 2, and 1r' is a square. This shows that the fixed p oi nt s of WH are rational. One can also show t hat the quat erni on algebra corresponding to Xo (N}P is given by (N- , p.). More generally!

Exercise: Let F be

a.

field, and let

X

be the variety obtained by twisting P I

by t he quadrati c character attached to t he exten6ion F( v'b) and the involution

P I = � 4/= . Then the quaternion invariant of this curve Br2 (F).

is

of

the element (4. b) of

The ramification in the Shih covering constructed above is as fo llows :

N==2:

is of type (2, p, p). One ramifica.tion p oi nt is rational over Q , with inertia group of order 2 generated by the element The two ot hers are defined over Q( H) and are conj u gate to each other. Their i nerti a group s, of order , . correspo nd to the two different conj u ga. cy classes of unipotent elements in PSL2( Z ), and where a is a non-quadratic residue mod p. The ramification

(�l 5)'

(A i )

(6 1)'

C hapter 5.

52

Elliptic curves

N =3: The situation is simi lar ; we have ramification of type (3J P, p ) , inertia group of or d er

N =1:

The

being generated by the element ( i (01 ) .

3

ramification is of ty pe

(3, 3, p, P) i

nates, the ramification points are located at

Exercise:

that

S how

the

replaced by PSL2 (Z/P"Z).

statement of tho 5. 1 .1

(�)

Assume as before that ---+

Xo(N'f

= -1.

and

±H.

remains true when

is

PSL2 (Fp)

the case where

r a t ional

Then there is a regular Galois covering

with Galois group G

theorem, one takes N such that

over

wi t h a suitable choice of coordi·

±�

A complement

5 .4 C

th e

Xo( N) is

PSL2 (Fp) O.

=

Xo( N)

h as genus

by the above.

of genus I j for if the twisted curve

point . an d is of rank > 0 (where

Xo(N)P

In Shih's exp loi t

One might try to

Xo(N)P

ha.s a Q­

is viewed as an elliptic curve

Q by fixing this rational point as origin) th en the following variant of

HilbertJs irreducibility theorem allows us to deduce the existence of infinitely

many extensions of Q with Galois group G:

Pr oposition 5.4.1 Let C

---+ E be a regula.r Galois covering with group G, where E is an elliptic curve over a number field K. A.uume tha.t for ev­

ery proper subgroup H of 0 conta.ining the commuta.tor 6ubgroup ( Ot G) J the corresponding G / H -covering of E is ramified IJ.t letlSt one point. Then a.ll P E E(K) ezcept a finite number ha.ve property Irr(P) 0/ §9.3.

lit

The hypothesis in the proposition

t he covering

C /H

coverings of E

are

Falt i ngs ' theorem,

---+

implies

abelian. Hence the genus of

CH

proposition.

P

s

CH

H

:I G,

ince the only unramified =

C/ H

is at least 2; by

has finitely many rational points . Let SK C E(K) be

the union of the ramification points and the

finite s e t , and if

that for all subgroups

E is rami fie d somewhere}

rt SK, then property

images of the CH(K). This is a is s at i s fied. This proves th e

Irr( P)

The above result was first obt ained by Neron (Proc. Int. Congo 1954 ) in

we ak e r

form, since Mordell's conjecture was still unproved at that t ime .

Corollary 5.4.2 Assume E( K) is infinite.

a

Then, there are infinitely many

linearly disjoint Galois extensions of K with Galois group G.

Proof' If L is any finite extension of K, we can find P E E(K ) , P rt SL (where SL is defined as above) . Th e property Irr(P) is then satisfied both over K an d over

L.

The corresponding G-extension Kp is then linearly disjoint from

The corollary follows.

L.

53

5 .5. Furths relults

For N = 1 1 , p = 47, the twisted elliptic curve XD(Nr has rank Q. Hence there are infinitely many extensions of Q wit h Galois group PSL:.(F4T)' ( An explici t example has been written down by N. Elkies. )

Example:

2 over

Problem:

Is it possible to generalize prop.

5 .4.1 to abelian varieties? More

pre­

cilely, let 11' : C - A be a. finite covering of an abelia.iJ. variety A. Aalume that a.) ACE) il Zarlw denle in Ai b) 11" is ramified (Le., C is not an abelian variety). Is it

trUe that 1I"(C( K» it "xnuch Ima.ller" than A(K), i.e., that the numbs of points of logarithmic height :::;; N in 1I'(C(K» is o( NPI2), where p =:: rank A{K)? There il a pi.4tial relult in this direction in the pa.per of A. N eron referred to above.

5.5

Further results on as Galois groups

Concerning the group. results: 1.

PSL2(F I )' for q

(b)

q

( d) 2.

a. prime, is

8, by a result

=

q = 25, b y

q = p1 ,

S L2 (F q

)

is isomo rphic to As and �

of Matzat «(Ma.3J ) .

a result of Pryzwara ([Pr]).

for p p rime ,

( al q = 2 , 4, 8,

and

finite fieldlt there are the following

PS�(F,)

p :; ±2

(mod 5), d.

is known to have property

SL:.(F,)

)

known to ha.ve property GalT when:

= 4 and q = 9, because respectively in these cases.

(a) (e)

over

PSL:a and SL2

q not

P S L 2 (F q

GalT

[Mel].

for:

for then

SL, ( F9) � P S L.. ( F, ). (b) q = 3, 5 or 9, for then SL:l (F'I) is ilomorphic: to .,4", As, or �, which are known to have property G alT1

( c)

q

=

7, Mestre (unpublished).

It seems that the other values of q case q ::: 11 ).

d.

ha.ve

[Me2}, or §9.3.

not been treated (not even the

of Galoil extenlions of Q with Galois group 1 , 2, . . . , 1 6 . Their conltruction is due to Mestre (unpub­ lilhed), who uses the representations of GQ given by modular forms mo d 2 of prime level :5 600. There are a. few exam p les

S�(FII-)

for

m. =

Chapter

6

Galois extensions of C (T) 6.1 Our

The GAGA principle

goal in this chapter is to construct G alois extensions of C(T),

us

ing the need

tools of topology and analytic g�metry. To carry out this program, we

a bridge between analysis and alge bra .

Theorem 6. 1 . 1 ( GAGA principle) Let X I Y he projective algehrtlie vtlneties over C J and let Xaa, yaa be tAe corresponding comple: tln4lytic spaces. Then. 1 . Every I1n41ytic mal" xaa --+ y'" is algebf'tJic. t. Every coherent antilytic sheaf over Xaa

mology coincides with W tJn4lytic one. For

a

is tdgebrtlic, tJnd its a/gebrtlic cohD­

proof, see [Sc4] or [SGAl] , expo l e XII. In what follows , we will allow

ourselves to write X instead of xaa .

Remarks: 1. The functor X

t-+

Xaa is the "forgetful" functor which embeds the category

of complex projective varieties into the category of complex analytic s paces .

Th. 6. 1 .1 implies that i t is fully faithful. 2 . By t he above, there is at most one algebraic structure on a compact analytic space which is compatible with it .

3.

Th. 6. 1 . 1 implies Chow's theorem:

every closed analytic subspace of

p roj ec ti ve algebraic variety is algebraic. 4. The analytic ma.p exp : G. --+ G., where

is not al geb raic; hence the hypothesis that X is projective is essential. Exercise;

If

isomorphism

a

X and Y are reduced varieties of dimension 1. prove tha.t any analytic of X on Y is algebraic; disprove this for non-reduced varieties. 55

Chapter 6. Galois

56 Theorem 6. 1 . 2 (Riemann) Any mension 1 is algebraic.

extensions of

C(T)

compact complez analytic manifold of

di­

The proof is easy, once one knows the finiteness of the (analytic) cohomology 2 . A generalization for a

groups of coherent sheaves, see e.g. [Fo] , chap.

broader class of varieties is given by a theorem of Kodaira:

Theorem 6. 1.3 (see e.g. [GH] , ch. I, §4) Every compact Kahler mani.fold X wh.o3'! Ki.hler cl4S3 is integral ( as an element of H2 (X, R)) is a projective algebraic variety. In the above theorems, the compactness assumpti on is essential. For coverings, no such assumption is necessary:

Theorem 6.1.4 Let X be an algebraic variety over C, and let 11'" : Y ---' X be a finite unramifietl. analytic covering of X . Th.en there is a unique algebraic 3tructU?"e on Y compatible with its analytic 3tructure and with 11'" . The proof i s given in [SGAl], expose XII. Let us explain the idea i n the case

X

=

X

- Zl

where X is an irreducible projective normal variety, and Z a

closed subspace. One first uses a theorem of Grauert and Remmert [G R] to

extend Y

--+

X to a ramified analytic covering Y

--+

X. Such a covering

corresponds to a coherent analytic sheaf of algebras over X. Since X is pro­

jective, one can apply tho 6 . 1 . 1 , and one then finds that this sheaf is algebraic,

hence so are Y and Y.

(The extension theorem of Grauert and Remmert used i n the proof is a rather delicate one - however, it is quit e easy to prove when dim X = 1 , which is the only case we shall need.)

The GAGA principle applies t o real projective algebraic varieties, i n the we may associate to any such variety X the pair (XAIl, s ) ,

following way: where

X'"

=

X (C) is the complex analytic space underlying X I and

is the anti-holomorphic involution given by complex conjugation on X. The real variety X can be recovered from the data (X-, s ) up to a unique iso­ morphism. Furthermore, any complex projective variety X together with an &

R: for by X to X I the conj ugate variety of X, and hen ce gives descent data which determines X as a variety over R.

anti.holomorphic involution GAG A ,

�

determines a projective variety over

is an algebraic isomorphism from

Simi la.r remarks apply to coherent sheaves, and coverings: for example, giving a finite co'Vering of

X

over

complex analytic space

R is equivalent to giving a finite covering of the

X(C),

together with an anti-holomorphic involution

which is compatible with the complex conjugation on

X(C).

57

6.2. Coveringl of Riemann lurfa.cel

6.2

Coverings of Riemann surfaces

Recall that for all 9 � 0, there exilts, up to homeomorphism, a unique c;om­ pad , mnnected, oriented lurface Xg of genus g (i.e., with first Betti number 2g ) . This can be proved either by "cutting and pasting" or by Morse theory (in the differentiable c;ategory). The surfac;e Xg has a standard description as 1 " 4" b" 4; 1 , b;l , and a polygon with 4g edges labelled 41. bt , 4 1 , bIl, identified in the appropriate manner. From this desaiption one can see that the fundamental group 1I'1(X,) relative to any base point has a presentation given by 2g generators at , 6 I , . . . , "" bg, and a single relation • •

1 41 6141-l b-l 1 ' " all bga,-l bg- =

1.

(To show this, one may express Xg as the union of two open sets U and V, where U is the polygon punctured at one point, which is homotopic to a wedge of 2g circle.and has fundamental group the free group F on 29 generators, and V is a disk in the center of the polygon. Applying the van Kampen theorem, One finds that 1I'1 (X,) = 1 *8 F F/(R), 1 0-1 L . R where IS 41"1 41 I • • • a, bg4g-1 6-1 , ' ) Let noW Pl . ' . . , Pit be distinct point. of X"� and let WI be the fundamental group of X, - {PI , . . " PI;}, relative to a base point x . E ach 11 defines in '1['1 a conj ugacy class Oi corresponding to "turning around p. in the positive direction" (choose a disk D. containing � and no other Pi ' and use the fact that 2I'"1 (Di -P,;) can be identified with Z). A similar argument as above proves that the group 1rl has a presentation given by 29 + Ie generators 41 1 bt I • • 'J 4" btl , Ch " CAl and the single relation: =

• •

where the c; belong to 01. for all i. We denote this group by 1rl (g, Ie). Observe that, if k � I , t hen 1I'1 (g, Ie) is the free group on 2 g + k - 1 generators. In the applications, we shall be mainly interested in the case 9 = 0, 1c � 3.

6.3

From C to

Q

Let X be an algebraically cloled field of charaderistic 0, X a projective smooth curve of genus 9 over X, and let Ph , . . , P, be distinct points in X(K). The algebraic fundament al group of X - {PI " ' . ' Pi} may be defined as follows. Let X(X) be an algebraic closure of the function field X(X) of X over K, and let n c X(X) be the maximal extension of K(X) unramified outside the

Chapter 6. Galois extensions of C(T)

58

points Ph " ' , Pic - The algebraic fundamental group is the Galois group of n over K(X), namely,

r;ta(X

-

It is a profinite group. Let

{Ph ' . . , Pic } ) us

=

Gal( n / K (X)).

denote it by 71" .

Theorem 6.3.1 The algebraic fundamental group 1f' is isomorphic to the profi­ nite completion of the topological fundamental group :

(Recall that the profinite completion r of a discrete group r is the inverse

limit of the finite quotients of

r. )

Proof: Let first E be a finite extension of K(X) with Galois group GJ let

Qi

be a place of E above Pi, and let I( Qi) denote the inertia subgroup of G at Q.. Since the ramification is tame, we ha.ve a canonical isomorphism:

where e!i is the ramification index at Q,; this isomorphism sends an element " of I(Qi) to .s7l"/'If (mod 71" ) , where 71" is any uniformizing element at Qi. From now on, let Q-i be a place of {} above Pi . The isomorphism a.bove can be generalized to such infinite extensions, by passing to the limit: write 51 as a union of finite extensions E; with ramification indices ej at Qi.. Then one has:

I( Q i)

�

�P.ei '

Choose a coherent system of roots of unity in K , i.e. , a fixed generator for each �, such that z:n � for all n, m. This provides I ( Qa) with a canonical generator c( Q .: ) . Changing Q. above P.: merely affects ce Q ,, ) by inner conjugation. Hence, the conjugacy class Ci of c( Q.:) depends only on Pi. (Changing the coherent system {� } - i.e. , replacing it by { z: } , where .. a E Z· Aut Z- replaces c(Q, ) by C(Qi)4.) Th. 6.3.1 is a consequence of the following more precise statement:

Zn

=

=

Theorem 6.3.2 There eZ'ists a choice of element3 :I;j, Yj, Ci in 71" I with Cw E Ci for All i, such thAt: 1 -1 - 1 - 1 -1 1 . XIYtXt !It x"YgXg Yg Ct ' " c} !. The map 7r't (gl k) --+ 71" mapping the generators fI!;, !ljJ Cw Of 7l" 1 (g, k) to the element3 X ii Yi, C. extends to an isomorphism • • •

=

.

6.3. From C to Q

59

In other words, r is presented as a profinite group by the generators

Ci

Zi, Yj,

with the relations above. The proof will be done in two stages: first, for K = Cl then, for arbitrary algebraically closed fields K of characteristic 0.

= C, with the standard choice of roots of unity, {%on} = {e'1ri/,, }. In this case, the result follows from the GAGA dictionary between algebraic and topological coverings (see tho 6.1.3 and [SGA1]).

Step 1: Proof when K Step

2:

K arbitrary. One has the following general result

Theorem 6.3.3 Let V be an algebraic variety o'Uer an alge.braically closed field K of characteristic 0, and let K' be an alge braically closed e:tension of K . Th.e.n any covering 0/ V over K' comes uniquely from a covering defined over K .

(Note that this is not true in characteristic p, unless the coverings are tame. For example, the Artin-Schreier covering yP - Y = aT of the affine line with a E K' » a f/. K, does not come from a covering defined over K.) Two coverings of V defined over K and K� -isomorphic are K�isomorphic; this is clear, e.g. by using a specialization argument. Next, one has to show that any covering which is defined over X' can be defined over K. II K' is of finite transcendence degree over K, then by using induction on tr.deg(K' I K), it is enough to show this for tr .deg( K' / K) = 1. In this case, a covering W --+ V over K' corresponds to a covering W x C --+ V x C over K, where C is the curve over K corresponding to the extension K' I K. This can be viewed as a family of coverings W --+ V over K parameterized by C. But it is a general fact that in characteristic 0, such families of coverings are constant. (One can deduce this from the similar geometrical statement over C, where it follows from the fact that rl (X X Y) = 71'1 (X) X rl (Y). ) By choosing a K-rational point c of C, on then gets a covering We --+ V defined over K which is K' .isomorphic to W.

Step 3: Using tho 6.3.3, we may replace ( K, {z.}) by (Kl , {z,.}), where tr.degKt /Q < 00. One can then find an embedding Kl --+ C which trans­ forms the {z.. } into the {e21ri/"n (irreducibility of the cyclotomic polynomials). Another application of tho 6.3.3 concludes the proof. Remark:

Let

r

be a discrete group, r its profinite completion

t

=

lim r/N -

where N runs over the normal subgroups of r of finite index. The canonical homomorphism r --+ r is universal for maps of r into profinite groups. It is not always true that this map is inje.ctive. There are examples of finitely p� sented r with r infinite but r {I}. e.g. the group defined by four generators Xi (i E Zj4Z) with the four relations XiZi+1X, 1 = z�+l (G. Higman). =

Chap te r

60

Problem:

Can IUch

6. Galoil extensions of C(T)

a r be the fundamental group of an algebraic variety over C?

In the case of curvel, which il the case we are mostly interested in, the map 7I' l ( g . k) --+ 1i\(g, k) is injective. This follows from the fact that 7I'l(g, k) is isomorphic to a subgroup of a linear group (e.g., S�(R) or even better, SL2(R)); one then uses a well-known result of Minkowski and Selberg (see e . g. [Bor}, p. 1 19).

6 .4

Appendix:

universal ramified coverings

of Riemann surfaces wit h signature Let X be a Riemann surface, S e X a finite set of points in X. Let us assign to each point P E 5 an integer np, with 2 ::;; np < 00. Such a set 5 along with a set of integers np is called a signature on X. One defines a ramified covering subordinate to the signature (5j np) to be a holomorphic map f : Y ---+ X. with the following properties: 1 . If Sy ;;;: /-1 (5), the map ! : Y - Sy ---+ X - S is a topological covering.

{Pl. Then f- l (D) splits into connected components Da , and the restriction fa of f to DQ is isomorphic to the map z ..... z..... , for some na dividing np.

2. Let P E S and choose a disk D in X with D n s

=

(When Y --+ X is a finite map, condition 2 just means that the ramification above P divides np .) For every P E S, let Cp be the conjugacy class "turning around P" in the fundamental group 'Kl (X - 5). If Y --+ X is as above, Y - f-1(5) ----+ X - S is a covering, and the action of c;r is trivial. Conversely, given a covering of X - 5 with that property, one shows (by an easy local argument ) that it extends uniquely to a map Y --+ X of the type above.

Theorem

6. 4 . 1 Let X, 5, and np b e given. Then there is a universal co ver­ ing Y subordinate to th.e signature (S i np) with. !a.ithju.l action of

where N is th.e normal subgroup genera.ted by the Y is simply connected} and one has Ylr X.

s"P I

s E CPI P E S. This

=

Sketch of proof. Let YN be the covering of X - 5 associated to N. This is a Galois covering. with Galois group 'K1(X - S)jN. As above, one can complete it to a covering Y ---+ X. One checks that Y is universal, simply connected, and that r acts properly on Y, with X = Y/r.

61

6.4. Appendix

We apply this to the case X = X T, where X is a compact Riemann surface of genus g, and T is a finite subset of X. Letting S = {Ph - . . , p. } , p. +th one associates to X a "generalized signature'll on and T = {p.+1 , X, where the indices are allowed to take to value 00 , by set.ting -

• • •

{ ?Ii

=

nPt

if i � a , otherwise.

7l.i = oo

The corresponding group r is then defined by generat.ors C'H and relations:

Cl ,

{

. • .

-1 b-1 1

41 bl(1t

c!tl

=

1,

. . .

•

• •

-1b-1 " CI '

Ie tlgl'IgG.

, c:.

=

1.

• •

'1+,

=

41 ,

bt ,

. . .

, Gil' b;,

1

Example: In the case 9 = 0 and a + t = 3, the group r and the corresponding universal covering X --+ P l (e) can be constructed. geometrically as follows. Set � = lint + 1 /n2 + 1/n3 (with the convention that 1/00 = 0). If � > 1, let X be the Riemann sphere S2 equipped with its metric of curvature 1; if l. = 1 , let X be C with the euclidean metric; and if � < 1, let X be the Poincare upper half plane, equipped with the metric of curvature -1. In each case, by the Gauss-Bonnet theorem there is a geodesic triangle on X with angles 1r/nh 1r/n2' and 1rln3 ' (The case f&.i = 00 gives an angle equal to 0, and the corresponding vertex is a "cusp" at infinity.) Denoting by 8; the reflection about the ith side (i = 1, 2, 3), the elements 01 = 8283. C2 = 838h and 03 = 8182 satisfy the relation

and the 0, are of order ni . Hence the group r' generated by the C.. is a quotient of r, by the map sending ca to C, (i = 1, 2, 3). In fact, one can show that the projection r --+ r' is an isomorphism, and that X with this action of r is the universal covering of tho 6.4.1. A fundamental domain for the action of r is given by the union of the geodesic triangle with one of its reftections about a side. Theorem 6.4.2 cases :

1. !.

9 =

9

=

OJ OJ

The element ca of r has order ?Ii, e2;cept in the follo1l1ing two

8 + t = 1; a

+t

=

2, "' 1 :f:.

n1 ·

Consider first the case 9 = O. This case can itself be divided into three subcases: 1. 8 + t = 2, and "' 1 = n2 . In this case r is isomorphic to a cyclic group of order Rl (if "'1 00, then r = Z), and the statement follows. =

62

Chapter 6. Galois extensionl of

2. & + t order n•.

=

3.

The explicit const ru ction of

r ab ove shows tha.t the

C(T)

Cs are of

4. One reduces this to case 2 by adding the rela.tions Ci = 1 for a of in dic es with III = & + t - 3. One thus get s a surjective homomorphism onto a group of the type treated in case 2, and the Ci with i rI. I are therefore of order 71i. By varying I, one obtains the result. 3.

a

s et I

+t�

To treat the case 9 � 1 , it suffices to ex hib it a group containing elements Al , . . . , A., Bl t . . . , B., C1, , C.+& satisfying the given rela.tion for r with the C, of order "i , (i = 1, . . . a + t). For this, one may choose elements Ci of order ns in the special orthogonal group SO J. Every element z of S02 can be written as a commutator (y, % ) in the orthogonal group OJ ; for, choosing y E S0 2 such that y2 =: :1:, and z E O2 - S 02, one has yzy- 1 %-1 Y . 'II = � . Hence, i t suffices t o take Aj, Bi = 1 if i > 1 , an d A I , Bl E O J such that • • •

I

=

that the signature (nt l . . . , n.H ) is not one of the exceptional cases above. We then have a "universal" ramified covering Y which is simply connected. By the uniformization t heorem (d. [Fo] ) , such a Y is isomorphic to either P l (O), Ot or the upper half plane ?t. To t ell which of these three cases occur, let us introduce the "Euler-Poincare characteri sti c" E of the signature: .+& 1 E = 2 - 2g - L(1 - - ) , Assume

in the theorem

with the convention

that

.1. l:1l:I

=

1ti

1=1

O.

Theorem 6.4.3 1. If E > O J !. If E = 0 ) 3. If E < O J

This can

1.

Case

1

then Y is isomorphic to Pl (e). then Y is isomorphic to C . then Y is isomorphic to the. upper half plane 11..

be proved using t he occurs only

if 9

G auss- Bo nn et

= 0 and the

•

empty, corresponding to the

•

(n, n), n

< 00, corresponding

•

(2, 2, n),

corresponding to

•

(2, 3, 3),

corresponding

•

formula.

signature is;

identity t o the

covering Pl (e)

cyclic

---+

Pl (C) .

covering Z 1-+ zft.

r = Dft , the dihedral group of order 2n.

to r

=

A ..

( tetrahedral

group).

(2, 3, 4), corresponding to r :::; 84 (octahedral group ).

63

6.4. Appendix •

(2, 3, 5), corresponding to r

=

A5 (icosahedral group).

2. Case 2 occurs for •

9

•

g =

=: 1 and empty signature, corresponding to the covering of a complex torus by the complex plane, r = a lattice.

0 and signature (00 , 00 ) , corresponding to the map exp : C

•

----.

r :: z.

C,

signature (2, 4, 4), corresponding to r ing the Ia.t tice Z + Zi.

=

group of affi ne motions preserv­

•

signature ( 2 , 3 , 6 ) , corresponding to r = group of affine motions preserv­ ing the lattice Z + C+p)Z.

•

signature (3, 3, 3), corresponding to a subgroup of index 2 of the previous one.

•

signature (2, 2, 2, 2), corresponding to coverings

where T is an elliptic cutve and C covering map.

�

T is the universal unramified

3. Case 3 corresponds to the case where r is a discrete subgroup of PSL2(R) which is a "Fuchsian group of the first kind" , cf. e.g. Shimura [Shim] , Chap. 1; one then has X :: 11./r and X is obtained by adding to X the set of cusps (mod r). (By a. t heore m of Siegel (S ieJ , such groups r can be cha.racterized as the discrete subgroups of PSL2 (R) with finite covolume.) The genera.tors Ci with n. < 00 are elliptic elements of the Fuchsian group r, the t!.j with 1J.i = 00 correspond to parabolic elements, and the tlj and b; to hyperbolic elements.

Chapt er 7 Rigidity and rat ionality on finit e groups 7.1

Rationality

Let G be a finite group, CI( G) the set of its conjugacy classes. Choose N such that every element of G has order dividing N; the group rN = (Z/NZ) · acts on G by sending s to s� , for a E (Z/NZ )" and acts similarly on Cl( G) Let X( G) be the set of irreducible chara.cters of G. They take values in Q(J&N) , and hence there is a natural action of Gal(Q(PN)) � rN on X(G). The actions of rN o n CI(G) and on X(G) are related by the formula .

O"Q (X)(S )

=

X ( sQ),

where U'Q E Gal(Q(J&N )/Q ) is the element sending the Nth roots of unity to their ath powers.

De6nition 7. 1 . 1 A class c in CI(G) is called Q-rntional if the following equivalent properties hold: 1. c is ji%ed under rN A !. Every X E X(G) ta.lces valu es in Q (and hence in Z) on c. (The equivalence of these two conditions follows from the formula above.) The rationality condition means that, if s E c, then all of the generators of the cyclic group < s > generated by s are in c, i.e., are conjugate to s. For instance, in the symmetric group Sn , every conjugacy class is rational. More generall y, let K be any field of characteristic zero. The class c of an element s E G is called K-rational if X( s) E K for all X E X( G), or equivalently, if c« = c for all a such that O'Q E Gal(K(PN)/ K). For example, the altern ating group As has five conjugacy classes of order 1, 2, 3, 5, and 5 respectively. Let us call 5A and 5B the two conjugacy classes of exponent 5. (They can be distinguished as follows: given a five-cycle 8, let 65

Chapter

66

7.

Rigidity and rationality on finite groups

a(s) be the permu t ation of { 1 , 2, 3, 4, 5} sending i to si(1). Then s is in the conjugacy diUS 5A if and only if a(s ) is even. ) If " E 5A, then S -1 E 5A and s2 , 83 E 5B, 80 the classes 5A and 5B a.re not rational over Q. However , they are rational over the qu adratic field Q( v'S).

Remark: The formula O'a (X)( .t ) = XC,CI) has an analogue for supercuspidal admis. sible representations of semi-simple p-adic groups. This can be deduced from the fact, proved by Deligne (d. [Del]), that the character of such a representation is supported by the elliptic elements (note that these are the only elements .s for which the notation "a makes sense). Rationality oC in ertia

Let K be a local field with residue field k of ch ara cterist i c zero, and let M be a Galois extension of K with group G. There is a unique maximal unramified ext ension L of K in Mj it is Galois over K. Let I be the inertia subgroup of G, i.e. , I = Gal(M/ L). Si nce the ramification is tame, I is cyclic. In fact] there is a canoni cal i somorphism

whe re e denotes the ramification index of L I K , and l is the residue field of L. (This isomorphism sends 0' E I to U1f 171' ( mo d 71' ) , where 1f is a u nifor mizing element of M.)

Proposition 7.1.2 Th.e ddSS in. G of an element of I is rational over /C . Indeed, the group H = Gal(LI K) = Gale'/ k) acts on P e e l) in a natural waYJ and on I by conjugation. These actions are compatible with the isomorp hi sm above. If now (1C1 is an element of Gal(k(p. )lk), there exists t E H such th at t acts on J'e (l) by z 1-+ zQ . If 8 E I, then sa and tst- 1 have the same image in Pe(l ). Hence SCi = ta r l . This shows that the class c of s is such that tf& = c, q.e.d. Corollary 7.1 .3 If Ie

=

Q, then. the cldS8es in I are rational in G .

Remark on the action oC rN on CleG) and X(C) The rN-sets C1(G) and X(G) are the character sets of the etale Q- alg eb ras Q ® R(G) and ZQ[G), where R(G) is the repre s ent ation ring of G (over Q), and ZQ[G] is the center of the group al ge br a Q[G]. These rN-sets are easily proved to be "weakly isomorphic" in the sen se of exercise I below . However, they are not always isomorphic (see [ThID. Exercise: 1.

Let X and Y be finite sets on which acts a finite group r. (a) Show the equivalence of the following properties:

7.2 . Counting solutions of equations in finite groups

67

::: Iye l for every cyclic subgroup C of r; jXjCI = IYICI for every cyclic: subgroup C of rj

i. I XC I

ii.

iii . l X /H I

==

IYIHl for every subgroup

H or r ;

iv. The Q-linear representations of r defined by X aJ).d Y are isomorphic.

If those properties are true, t he

(b)

Show that

r·sets X and Y are said to be weakly isomorphic. weak isomorphism is equivalent to isomorphism wh en r is cyclic

( "Brauer's Lemmaft ). Give an example where r is a non-cyclic group of order I X I = I YI ;::: 6, and X and Y are weakly isomorphic but not isomorphic. 2. Use exerc. l o(b) to prove that, isomorphic to Z Q[ G] . 3. Extend prop. 7. 1.2 to

MIX 7.2

is tame.

if G is

the case where

Ie

a

p-group, p

-F 2,

has cha.racteristic

then

l' >

Q ® R(G)

4, is

0, assuming that

Counting solutions of equations in finite

groups

G be a compact grouPi e quipped with its Haar measure of total mas s 1. Let be an i rred ucible cha.racter of G, wi t h p : G ---+ GL( E) the c orresp ondi n g X linea.r representation. By Schur's lemma , IG p(tzt- I ) dt is & scal ar multiple l · I E of the identity in GL(E). Taking traces gives xCz) = lX ( l ), hence

Let

X ( p(tzt-l)dt = ( x ) lB' JG XCI ) Multiplying on the right by

p(y ), we get :

k p(tz,- 'y)dt ����p(y). =

Taking traces

give s :

X( ) (y ) .

I X(txt-1y) dt == 2: X ,,( 1 ) Je This formula extends by induction to k elements Xl ,

• •

'J 2: 1: :

(7. 1)

where ex

is the inne r pro d uct of ,; with the irreduci ble character X , ex =

fc ,;(x )X(x ) tb:,

68 and

Chapter assume that

E l ex lx( 1 )

summi ng over all chara.cters

is equal

to

<

7.

Rigidity and rationality on finite groups

so t ha.t

00

the

sum is normally convergent. By see that the i ntegral

in formula 7 . 1 J we

'"

't Cx

X(Zl) " . X(Zk)X(Y) . X( I )1&

Let us compute 1(¢) in the cue where G is a finite gro up and rP is the Dirac is 1 a.t the identity element J and 0 elsewhere. One has

function which

1

t/J and hence

ex

=

I(¢) is N/IG ,",

x(l)/ tGI.

where

=

I GI

� X(I)X,

, ZJu and 11 in G, the value of number of so lut ions (th . . . I t.) of the equation

Given elements Z1 ,

N is

the

• • .

Hence

(7.2) Let 01 , , 0" denote the conjugacy c1a.sses of the elements % 1 t d efine nCOl, " . , e.) to be the number of solutions (91 , . . . , 9,,) of the • • •

• • •

9192 ' " 9" Letting Z, be the order

=

I,

and

9i E Ca.

of the centralizer of an element of Ci t one has

By applying formula 7.2 and ob serving that Z, Theorem 7.2 . 1 The number n

91 ' " g. = 1 , 9" E Oil

I %. ,

equation

is given by

=

= nCGl , . . . 1 C� )

IGI/ IO,:! , one therefore get s : of solution.! of the equation

I I C1 , . . . ' C I � X( z d · · · x(z .) n= _ I GI

" �

X( I ) I:-2

'

'Where :1::, is a representative of the conjugacy class Gil (Inti the in-e4'Ucible chara.cters of G.

X

"'118

through all

7.2.

Counting solution. of equations in finite groups

69

Exercises: 1. Let G be

a compact group and let p be an irreducible representation of G with character X. Show that 1 p(t:tt- Z- 1 )= tit = 1/x(�)2.

JJ

that the left hand side is an intertwining operator for p, and hence a sc:ala.r by Schur's lemma.. Then c:ompute its trace.) Conclude tha.t

(Hint: Show

J J p(tzr1z-1 y)dz p(Y)/x(1)2, for J J x(tzt-1 z-1 . y)ck dt X(y)/x( 1)2 . d1

•

and that

'Y E

all

=

is

G.

=

Hence show by induction that:

J . . . J x(tlzltl1z11 . . . t,z,t;1z;1 . y)dz1 dtl

• .

· dz, dt,

=

X{Y}I'X,<.1)2'.

2. S uppose now that G is finite. We denote the commutator "'11U- 1 .,, - 1 by (u, 'U ) . Let 9 � 0, and let 2; be fixed elements of G, for ; = 1, , k.

let N

=

Neg, �;. y ) denote the number of tuples of elements of G,

• . .

IUch that Using

( U,1 . 11t) . . . (U" 'U, )t l�lti'l . . . t 1 2.t. 1 exerc. 1, show that :

N

=

=

of u and 11 For y e G ,

,-1 ,

IGI2.+1l- 1 E x(Zl ) " ·x(J.)X(y)/x(lr �·+.It - I . x

3. Show that an element y of G is sum

a

produc:t of 9 commutators

if

and only if the

y / ( ) 1fL x( ) x 1 2 1 X

is non-zero. In partkular, 'JI is a. commut ator if and only if E X( y) / x(l ) :1 O. It is a knQwn Call sequence of the "classifica.tion theorem" that every element in a finite non·abelian simple group is a commutator. The reader ma.y wish to verify this (for one of the sporadic groups, say) by using the formula above and the character t ables in [ATLAS].

Remark: Th. 7.2.1 can be used to co mpute the number of subgroups in a. fin ite group which are isomorp hic to the alternating group As . Indeed, As has a prese nt a.tion given by three generator s z, y, z, and relations 2:2

=

1l

=

Z5 =

zyz =

1.

The problem therefore amounts to finding the number of solutions of the equa­ tion zy z = 1 , where z, y % belong to conjugacy classcs of exponent 2, 3, and 5 respectively. The same rem ark ap plies to Sol . At , and the dihedral groups, which have similar present at ions , d. tho 6.4.3, case 1.

Chapter 7. lligidity and rationality on finite groups

70

7.3 Let

Rigidity of a family of conj ugacy clas ses

G

be a fini te group, and fix co nj ug acy classes

Cl ,

.."

Ok in G.

Let

E = E ( Cl , . . . , CII:) be the set of all (gl J . . . , gll:), with g, E Ci, such that gl • • • gil: == 1. ( Hence, by the previo us section, n(el ) • • • J C,) = Itl . ) Let E = E(C1, , C,,) be the set of (gl � " " , gle) in E s uch that 91 , gle gener at e the group G. The gro up G acts by conjugation on E and t. • • •

' " "J

AI8ume tha.t the center of G is t rivial ( as is the case, for example, when G = S,,, n � 3, or when G is a non-abelian simple group). Then tkt: action of G on E is free: for, if 9 E G fixes (91 , . . . , g,,), it commutes with the g/s, and hence with all of G, since the gi 'a gener ate G; but then 9 = 1 , because G is assumed to have trivial center. One says that a k-tuple of conju gacy classes (C1 , " , GIe) is rigid if E # 0 • •

and

G

acts transitively on E, i.e. ,

One says that

(C1 ,

The order of

E

• • •

I

C. )

lEI

=

is strictly

\ G \.

rigid if it

is rigid, and

E ::;;; t.

can be comput ed usin g the formula of the p revi ous sec ti o n :

where Ci E Ci for i = 1, . . . , Ie, and X runs thro u gh the irr ed uci b le characters G. If Zi is the order of the centralizer Z(Ci ), t his can also be wri tten as:

of

(7.3) Rigi d ity is often pro ved in two steps:

1 . Compute the order of E, by us ing formula 7.3 and t he character table of C. 2. Compute the order of E - EJ by finding Ie-tuples i n E wh i ch do not gen· er ate all of G: for this, one uses a knowledge of the maximal subgro ups of G

(whenever possible).

Remarks: 1. Let

t7

be an outer automorphism

k-tuple of conjugacy cl as ses . Then that

(l

preserves each

follows that

Ci.

(Ugl , . ' " ug. )

inner conj ugation,

Then,

9

E

• • •

letting (g1 l ' " ' I gle )

b elo ngs to

there exists

of G. S upp ose (G1 1 , Ck ) is a ri gid for some i. Indeed, suppose

u(C.) I Ci

�. Sinc e

G such that

be an elem ent of E,

G acts transi tively on

for all i.

E

it

by

7.3.

Rigidity

of a family of conjugacy classes

71

But

the 9i genera.te G and hence u is an inner automorphismt which contradicts the assumption. 2. In many cases, the term E" x(Cl ) · · · X(C.. )/X(1 )1I- 2 in formula 7.3 is not

very large, the main contribution to the sum being given by the unit character X = 1. For a. rigid family of conjugacy classes, one might therefore expect that the order of magnitUde of I C1 1 · . - I Cil is clo se to I GI. As in §7. 1 , let a E (Z/N Z ) · , where N is a multiple of the exponent of G, so that (Z/NZ)· acts on Cl( G). Then one has: Proposition 7.3. 1 1 . It(Cf, · - . , Or) 1 = I E ( Cl , . . . J Ck ) !; 2 . I E( O�., . . . , 0:)1 := IE(C!, . . . , C. )l.

The first identity follows from formula 1.3 above, combined with the formula X(c") = O"m(X)(C) of §7.1 . The second is proved by i ndu cti on on the order of G, as follows: if H is a subgroup of G, let E{H)(C1 n H" " , 0", n H) denote the set of (g1 , ' " , 9i), with 9, E Oi n H for all it s uch that 91 ' " 91c = 1 and the 9i generate H. (In general, the C, n H are not conjugacy classes in H, but unions of conjugacy classes.) The formula

-

E - E( 01 ,

0 0 0

, 0",) -

U

HCG, H';:G

rJH) (0

L'

1 n H, . . . C. n H) J

supplies the induction step. Corollary 1.3. 2 11 (01 , ( Oi, . 0 , 0: ) .

• • •

, 011) iI rigid (respo

strictly rigiJ)1 so is

the ja.milll

0

Remark: Here is another way to prove prop. 1.3.1. Let F be the group with presentation given by generators Zl , 0, X. and rela.tion X l ' x. = 1 , and let F be its profinite completion. Then one has: . •

Proposition 1.3.3 For that

8( Xi )

is an automorphism (J 01 F 8Uch of xi . (Equiva.lently, there exist conjugate to zr, satisfying Yl . . . Y'" = 1, and gen­

each

Q

E

Z· J

• •

there

belongs to the conjugacy clo.ss

elements Yl, . . . , Y. with Yi erating F. )

Let us use the interpreta.tion of F as the algebraic fundamental group 7 of the projective line Pi with k points removed. By choosing a. coherent system {z,.} of roo ts of unity in Q, one has an identification (d. §6.3)

Chapter 7. Rigidity

72 Replacing the system

{z,.}

on finite g roups

and rationality

by {.;: } gives a different is omorphism, and com­

posing the two yields the desired automorphism 8 of

F.

1.3.3 implies that 1 � (Oi, . . . , 0:) 1 I E(OI, . . . , CJ: ) I . For, the el­ E(Ol , . . . , Ok) are in one-one c;orrespondence with the surj ective: homomorphisms F --+ G sending each 2:, to an element of Ci" while the ele­ ments of � ( ai , . . . , Or) correspond to the surjective homomorphisms sending each :1:, to an element in Cr. The automorphism (J : F --+ F induces a map Hom(F, G) --+ Hom( P, G) which gives a bijection between the two sets �(Cl1 " " 01:) and E(Oi, . . . , Of). Prop.

=

ements of

, C1:) i s rigid (resp. IItrictly Exercise: Show that , if (C1, (0.,.(1), . . . , O"'(J:» for any permutation q of { l , . . . , k}. • • •

family

7.4

rigid),

then so i s the

Examples of rigidity

We give only

a few such examples. For more, the reader should consult [Ma3}

and the papers quoted there.

7.4.1

The

symmetric group Sn Sn (n � 3) has cycles of order n, 2,

The symmetric group corresponding to

(nA, 2A, C(l ) is strictly rigid.

a cyclic arrangement of

conjugacy classes and

n- 1

respectively.

For, giving an n- cycle

{ l , . . . , n}

nA, 2A,

:t

E

nA

and

C( l )

The triple

determines

(i.e. , an oriented n-gon). Composi ng this

(n - i )- cycle if and only if the two vertices which are permuted are consecutive. Hence, the solutions

permutation of order

y, .;

.2: ,

n

of the equation

with a transposition gives an

:1:y,;

=

1,

with

:1:, Y, .;

cycles of order

n, 2

and

n- 1

respectively, are in one to one correspondence with the oriented n-gons with a

distinguished edge. Any two such configurations can be tranformed into one

another by a unique permutation in

(12)

and

( 1 2 . . . n)

strictly rigid.

Sn i

are known to generate

hence

Sn;

lEI

=

IGI.

More generally, c;onsider the conj ugacy dasses

nA, 2A ,

is the class of the permutation

(1 . . . As

b

efore,

an element

( :1:, y, .; )

E

k)( k

But E

this shows that

and

=

E, since

(nA, 2A, 0( 1 )

a(lr),

where

is

C (k )

+ 1 . . . n).

E c;orresponds to

two distinguished vertices separated by k edges.

an

oriented n-gon with

1£ Ie #- n/2,

any two such configurations can be transformed into one another by a uni que permutation in Sn, and hence l E I = IGI. However, (nA, 2A, a(l!l) is not rigid in generalj to get rigidity, one must assume that (k, n )

=

1 . In that case, any triple (z, Y J .; )

7.4. Examples of rilidity

73

in t generates 871: by relabelling if necessary, we may write :z: = (1 n) , and 11 = (1 1 k + 1 ) . Since ( Te, n ) I, the permutation zIc is still an n-cycle. By rel abelling again, the group generated by :z:1I and y is isomorphic to the group generated by the permutations ( 1 n) and ( 12). This in turn contains 1 , n) , which i s equal to 871 by a the group generated by (12), (23) , " ' 1 (n . . .

=

. . .

well-known result

-

(d. §4.4).

Exercise: Check th a.t It I = nl for the conjugacy classes (2A. nAt e(l» by applying formula 7.3 of §7.3 ( prove that the oD1y non-zero terms came from the two characters of degree 1 of 571), 7.4 2 ..

The alternating group As

The alterna.ting group As has unique conjugacy classes of order 2 and 3, de­ noted by 2A and 3A respectively. It has two conjugacy classes of order 5 which are ra.tional over Q( VS) and conjugate to each other, denoted by SA and SB.

P roposition 7.4.1 The fDllowing triples 0/ conjugacy cla..sses are strictly n... git! : (2A, 3At 5A) . (2A, 5A, 5B), a.nJ. (3A, 5At 5B). To prove this, one c an compute the order of E i n each case from the character ta.ble of AI ' In ATLAS style, it is: chara.cters

!

Xl )(2

)(3

X4 XS

60

4

lA 1

2A 1

3 3A 1

3

-1

0

3 4 5

-1

0

1

-1

0

1

5 SA 1 z' •

orders of f-centralizers 4- classes

5 5B 1 �

}I

I

-1

-1

0

0

One then gets:

1I:( 2A, 3A, 5A)1 ;;;

-

IE(3A , 5A , 5B) 1

+

602

-

IE( 2A , 5A , 5B) 1

60:1

. . (1 + 0 4 3 5

=

=

0 + 0 + 0) ;;; 60

1 1 . . (1 + 3 + 3 4 5 5 6 02

. . (1 + 0 3 5 5

+

+

0 + 0)

1 0 + 4 + 0)

=

=

60

60.

One checks easily that any triple in any of these E genera.tes follows.

7.4. 1

As, and prop.

Chapter 7.

74

Rigidity

and

rat ionality on finite groups

Exercises:

1 . S how that the t riples (3A, 3A, SA), (3A, 5A , 5A ) , and (5A, 5A, 5A) are strictly rigid. 2. ShoW' t h at (2A, 2A, 5A ) is not rigid, even though l E I 60 in that case ( the triples in E generate dihedral subgroups of order 10). =

7.4.3

The

groups

PSL2(Fp)

The group PSL2 (F,, ) , with p > 2, co nt ai ns unique conj uga.cy classes of ele­ ments of order 2 and 3, denoted by 2A and 3A respectively. There are two classes pA and pB of elements of order p, which are represented by u nip ot ent ma.trices, 1 and where -1.

(� l)

Proposition

(5 1),

7.4.2

One checks that

(�)

=

The triple (2A, 3A, pA) is strictly rigid.

(�oI Yo, zo)

E

E , where �o ,

matrices: yo =

1/0 . and Zo

(0 -1 ) 1 -1

are represented by

Zo -

_

'

(10 11 )

the

Conversely, let (�, Y, Z) be in E. We lift �, Y , Z to i, y, z in SL2(F,,), with i of order 4, y of order 3 and z of order p, So that we have zyz ± l . We vie'll these element s iUI automorphisms of the vector space V = F" E9 Fp . Let D b e the line of V fixed by z and let D ' iD be its tr ansfor m by i. O n e has D' :1 D (otherwise1 ±ii would not b e of order 3). After conj ugat in g by an ' element of SL2(F p ) , we may assume that D (res p . D ) is the first (resp. the second ) axis of coordina.tes in V. This mean s that we have =

=

-

:t =

for some A, I-'

in

F;.

( 0 -A ) A-I 0 '

By assumption,

and

i

=

(A i ) ,

z belongs to the

clus

(� l )i this ( () JI�l ) ,

pA of

we that p. i s a square. If we write p. = Jl 2 , and conjugate by see that we can further iUl sume that p. :;; 1, i.e., that i :;; zo o Moreover, since ii is of order 3 or 6, we have Tr (ii) = ± 1 . T his gives A = ± 1 , hence � = Xo, Y = 1/0 , and z = Zo, which proves the r es u lt .

impli es

Proposition 7.4.3 The triple (2A, pA, pB ) is strictly rigid if One checks that

(�Ol YO t ZO) is in E, where

(1 -1 )

�o = 2 -1 '

Yo - ( 01 11 ) ' _

(�)

=

-1 .

7 .4.

Examples of rigidity

The element

%0

75

is conjugate to

( A i) ; since (!) =

-

1 , it belongs to the class

pB . ConverselYt let (:tt Y, z) E t. We lift (:tt 1/ , z) &S a.bove to (f, y, z) with z of order 4 and j, j of order p. Let D be the line fixed by y and D' be the line fixed by i. We may again assume tha.t these lines are the standard coordinate lines, and tha.t

(1 1)

( )

z- = .\1 01 . Writing that jz has order 4, one gets Tr (ji) 0, i.e . .\ -

y == 0 1

I

=

=

- 2, q.e.d.

Proposition 7.4.4 The triple (3A, pA, pB) is strictly rigid if

(; )

=

-1 .

One checks tha.t (:to, Yo , zo) is in �t where

(

)

-1 :t o = 31 2 t

Yo =

(A 1 ) ,

The assumption tha.t 3 is not a quadra.tic residue mod p ensures that Zo is in the class pB. Conversely, let (:t, y, z) be in t. Using liftings (i, j, i) as above, one may &ssume that

( )

- (1 )

11 z = ).. 01 • 1/ == 0 1 ' Writing that jz has order 3 or 6, one gets Tr (ji) = ±1 , i.e. , ).. = - l or ).. = -3. However, ).. == -1 is impossible (it would imply that i belongs to the class pAl; hence .\ -3, q.e.d. =

7.4.4

The group SL2(Fa)

The simple group G = SL2(F s ), of order 504, h as three distinct conjug;u:y classes of order g, denoted 9A, 9B, and g O which are rational over the cubic field Q(cos 2;) and conjugate to each other (d. [ATLAS], p 6 )

. .

Prop osition 7.4.5 The triple (9A, 9B, 90) is strictly rigid. The character table for G is ([ATLAS], loco cit. ):

76

7.

Chapter

Xl X2 Xa X4 X.

504 lA 1 7 7 7 7

X7 Xa X9

9 9 9

X6

m

11

2A 1 -1 -1 -1 0

=

1

1 1

-2 1 1 1 -1

1

0 0 0

Y 11 Y :z:'

-2 cos 2; , 2. 2 cos 7 '

11'

7

7

7A 1 0 0 0 0

1

-1

8

=

9 3A

8

=

=

Rlgidity and rationality

7

7C 1

7B 1 0 0 0 0 1 Y Y.. y

0 0 0 0

9 9A

9 9B

1 1

1 1

:z:

z

:z:

:z:

x

:c"

1

-1 0 0 0

-1 0 0 0

Y Y

y

-2 cos � 9 , 2 coa !!' 1 ,

11 "

m" =


9 9C 1

on finite

groups

orders of +-- centralizers +-- cl as ses

1

z

"

z

Z

-1 0 0 0 ' " mm m

-2 cos a. 9, 2 cos l a,.. '

=

111J '11

"

=

=

1; 1.

Using formula (7.3) of §7.3, one gets: 1 �(9A , 9B , 9C) 1

==

5042

1

""93 ( 1 + "7

+

1

1

-;; + "7

+

1

"7

1

-

"8 + 0 + 0 + 0)

=

504

=

IGI·

Hence it suffices to show that any ( m, 1/, .z ) E E generates G to prove rigidity. The only maximal subgroups of G containing an element of order 9 are the normalizers of the non�8plit Cartan subgroups, which ate isomorphic to a semi�direct product C:z(F64 )1 , where (F64)1 denotes the multiplicative group of elements of F64 of norm l over F s , and the non�trlvial element of C'I. acts on (F;" )l by z � Z- l j they are dihedral groups of order 1 8. If ( x,y, .2 ) E E does not generate G, then m, y, .z are contained in such a normalizer. It follows that (by interchanging y and z. if necessary) : z =

But then :z:yz. proof.

=

:cl±H4

is

not equal to

1.

m%4.

This contra.diction completes the

Exercises:

1. Show that t he triples (7 A, 7 A, 7A), (2A, 3A , 7A ) , (2A, 3A, 9A) and that (7 A, 7 B I 7C) is not rigid. 2.

Let

G·3

be the automorphism

group of SL2 (FB), cr.

are s t

ri c tly

rigid ,

[ATLAS}, p. 6. Show t hat

(9A, 3B t 3C) is strictly rigidi t he class 9A is r at ional (as a class of G · 3); the classes 3B and 3C are rational over Q( �), and c onj uga.t e to ea.ch ot her.

the

triple

7.4. Examples of rigidity

7.4.5

77

The Janko group J1

The sporadic simple group

JI

discovered by Janko is of order

175560

=

� . 3 · 5 · 7 · 1 1 · 19.

2A, 5At and 5B of orders 2.. 5, and 5; the classes SA and 5B are rational over Q( v'S) and conjuga.te to ea.ch other. If :J: E SA, then Z- I E SAt but Z2 , z3 E S B ; these conjugacy classes behave like the ones of the same order in As.

It contains conjugacy classes

P roposition 7. 4.6 (d.

[Ho)) The trip le

rigid.

(2A, SA, 5B)

The relevant part of the character ta.ble of 175560

1A

1

Xl

""

120 2A

30

SA

30 5B

1

1

1

XS

76

-4

X7

S

1 1 2

77

-3

-z' -z

76

X6

4

77

Xa

77

-3

Xu

133

5

133

-3

%

Xu

133 209

-3 1

-1

Xu

Xa

-2 %

G

=

JI

is

rigid but nDt strictly

is (d. [ATLAS] . p.36):

orders of +-centralisers

of-

classes

1

1

2

z=

-z - z'

z'

(1

+

20

+

=

\1'5)/2

(1 - ,;5)/2

-2 Z

%

-1

Using formula. (7.3) of § 7.3, one obtains:

I�I � Hence

::;:;

'i2Q.iiJ" (1

17!5602

438900

-:;;

(2A, SA, 5B)

+ "

=

" + 20 +

7i - 7i

tiGI.

77

3

77

+

3

77

+

iii

3

133

+

3

i3i

+

I

20i

)

i s not strictly rigid. One can check that the triples in

E - E genera.te subroups of JI isomorphic to As. It is known that J1 contains

2 conjugacy

classes of such subgroups:

1. There are

IJ1 1/(21As l)

conjugate subgroups isomorphic to

contained in the centralizer of an involution in

J1•

As which are J) was first

(Indeed,

defined abstractly by Janko as a. simple group ha.ving the properly th at

it contains

an

involution whose centralizer

is isomorphic to

2. There is a conjuga.cy class of As·subgroups which are

there

are

J JI I/I As t

such subgroups.

{±l} X

As.)

self-normalising:

Chapter

78 In all, one has

� 1 J1 1/ 1A5 1

7.

Rigidity and rationality on finite

subgroup s of J1 which are isomorphic to

the conjugacy classes (2A, SA, SB) solutions in E. This shows that

are

-

rigid in

As,

each

subgroup

groups

As . Since gives I A5 1

3

I E - E I = "2 1 J1 1, and hence I � I

7.4.6

I J1 r,

=

The

i.e. ,

(2A, SA, SB) is rigid.

Hall-Janko group J2

604800 = 2733537. It has a r at io nal class 7A of order 7, and t wo conjugate classes SA and SB of order S, rational over Q( v'5), see e.g. [ATLAS), pp.42-43.

This sporadic simple group has order

Prop osition

7.4.7 (d. [HoD The triple (SA, SB J 7 A)

is

No proper s ub group of J'l has order divisible by 35, the other hand, formula (7.3) of j7.3 and the character loc.cit.) give:

strictly rigid..

lEI l E I . On of J:z ( [ATLAS],

hence table

=

Hence the result .

7.4 .7

The Fischer-Griess Monster

M

The Fischer-Griess group M, known as the "Monster" , sporadic simple groups. Its order is

is

the largest

of the

2 46 . 3 30 • S9 . 'f' - 112 . 133 · 1 7 · 19 · 23 · 29 · 31 · 41 · 47 · S9 · 71 , yet it has only 194 conjuga.cy classes. Its character table is therefore of man­ ageable size (in fa.ct , it was computed before M had been sh own to exist ) . The group M contains rational conjugacy classes 2A, 3B, and 29A of exponent 2,

3, and 29 (ATLAS

nota.tion).

7.4.8 (Thompson, d. [Hunt], [ Ma3] , [Th2]) triple (2A, 3B, 29A) is strictly rigid.

Proposition The

It can be verified by computer that

lEI

=

I M I . To prove

rigidity,

one must

no (z, Y J .z ) E E gene ra.te a proper subgroup of M, the maxima.l subgroups of M are not completely known at present . Hence, one must take the following indirect approach: suppose there

show that E Unfortunately,

=

f:, i.e

.

•

79

7.4. Examples of rigidity

a proper s ubgroup G in M which is generated by (z, 11, .c:: ) E :E. Let S be a simple quoti en t of G. Clearly the eleme nts z, 11, z have non-trivial image in 8, and hence the order of S is divisible by 2 · 3 · 29. Hence, it suffices to check that there are no simple groups S with 2 3 · 29 dividing l S I and l SI dividing IM I , such that S is generated by elements z, y, z conUng from the conj ugacy classes 2A, 3 E , and 29A in the Monster. This is done by checking that no group in the list. of finite simple groups satisfies these properties. One is thus forced to invoke the classification theorem for the finite simple groups to prove rigidity in this case. [Although the proof of the classification theorem has been announced, described} and a.dvertised since 1980, it is not yet clear whether it is complete or not� the part is

.

on "quasi-thin" groupa has never been published.]

Chapter 8 C onstruction of G alois ext ensions

of Q { T)

by the rigidity Ill e thod 8.1

The main theorem

Let K be a field of characteristic zero, let PI , " " Pk be distinct K -r ational points of P1 , and let OJ , . ' . , Ole be a family of conju gacy classes of a finite group G with tri v ial center. The following result is due to Belyi, Fried, Matzat, Shih, and Th ompson ( lee [Ma3] and [Se8] for references). Theorem 8.1.1 Assume th4t the f4mily (01 , . . . , 0.) is rigid and tlut.t eAch. Ci is rational. Then. there is a. regulAr G-covenn.g 0 --. PI defin e d over K which

is

un.ramified outside {PI " "

t

Pk} 4nd such that the inertia. group over

each P, is gen.eroted by 41'1 element of Oi. . uniqu e, up to a u1'lique G-uomorphism. By

taking K

=

Q,

one

Furthermore, such a coverin.g is

has:

Corollary 8. 1.2 G has property GalT (and hence is a Galois group over Q).

Proof of tho 8. 1 .1 Let L be the maximal extension of

K(T) unramified outside { Ph " " Pc } , the Galois group of L over K(T). This is the algebraic fundamental group of PI - {PI , . . . , PIli:} over fe (It is also called the geometric fundamental group b ecause the ground field is algebraically closed. ) Since L is iii. Galois extenlion of K ( T ) , one has an exact sequence:

and let

71'"

denote

1

--+ 11' --+ 1I'K -+

81

r

-+

1,

( 8 .1)

Chapter

82

8.

The rigidity m.et hod

where 1fK is the Galois group of L over K(T), and r Gal(K / K). Let I, be the inertia group of "II" at Pi. . As a profinite group, 11' has a presen­ tation given by Ie generators :1;1 , " x� and a single relation :1:1 ' :1:1: :::: 1, ct. tho 6.3 . 1 . More precilely. choose a coherent system { Za} of roota of unity in Q. This choice determines an element x, in each 1, up to conjugacy in 11' . One can then choose the Xi so that they satilfy the relation Xl ' " ZI: = 1 (cf. §6.3) . The set Hom( 11', G) of continuous homomorphisms 11' --+ G il equipped with natural G· and lI'K-actions. The G- act i on is defined (on the left) by =

• •

(g . f)( x )

=

.

9 E

gf( X )g-l ,

Gf

and the 11' x-action is defined (on the right) by (/ . a ) ( x )

=

f( a:z:a.- 1 ) ,

E

a

•

/ E Hom(1f , G),

/ E Hom(lI', G ) .

WK,

The two actions commute, i.e.t (g • f) • Q Consider the set H

C

=9

• (f • Q) .

Hom( 'If , G) defined by:

H = {,preP is surjective and ,p(:z:,)

E

Oi for all i} .

This set is stable under both the G and wK-adions: •

The action of G on itself by inner a.utomorphisms stabilizes the ai, and hence G preserves H , which is isomorphic to E(Oh ' ' ' ' Ok ) as a G-set (d. §7.3). By the rigidity assumption, G acts freely and transitively on H.

•

Conjugation by an element u E 7rK sends an inertia. group Ii at � to an inertia group at p{ . Since the Pi are K-rationa.I, Wx permutes the inertia groups above � . Hence, u sends each of t he Xi E Ii to a conjugate of :z:i, for some a E Z (namely, a = X ( u ) , where X is the cyclotomic character) . By the rationality of the C. , it follows that J . u maps each z, to an element of Cit and hence f . u E H.

Any � E H defines a. Galois extension E of K ( T) with Galois group G. To descend from K (T) to K(T), it suffices to show that ,p can be extended to '1CK. This is an i mmedi ate consequence of the following: Lemma 8 . 1 .3 Let 1

--+

A

--+

B

--+

a

--+

1 be

411. exact sequenet:

0/ Hom( A, G) as above ( i. e .J (g . f)(:z:) == gJ(x)g-l if 9 E G , :z: E A, and (! • b)(z) f ( b:z: b - 1 ) i/ b E B � :z: E A) . 1/ H is a non.-- empty subset 0/ Hom(A, G) on which G acts groups, let

G

b e 4 finite group, 4nd let

G

and B act on

=

freely and transitively, then the follo'Wing are equiv41ent :

,p E H tzttnds u.niquely to a homomorphism B H is stable u.nder the action of B .

1. A ny

I.

--+

G.

8.2. Two variants

83

1 holds , i.e. , any ,p E H ext en ds uniquely to a homomorphism G. If b E B, then (,p . b)(%) = �(b:tb-l) = tP(lnb- 1 ) (�(b) * �)(z). Hence � 'II b E H, since by hypothesis H is preserved by the action of G. Conversely, if property 2 is satisfied, then given � : A ---+ G, one may define ..p : B --+ G by: ,p * b = ,pCb) * ,p.

Proof: Suppose

..p : B

-+

=

SuCh a 1/1 exists (G acts transitively on H) and is unique (G acts freely on H). One verifies that ", defines a homomorphism B ---+ G w hi ch extends t/J; this follows from the compatibility of t he G- and B-actions. This completes the proof of the lemma, and hence of tho 8.1.l.

Alte",4de method of proof lor tho 8. 1 . 1:

(a) Prove that the required G-covering 0 ---to Pi exis t s over K and is unique up to a unique illI omorphism. (b) Use Weil's descent. criterion ( [Se3] , chap. V, no. 20) to prove that 0, together with the action of G} can be defined over K. (General principle: every "problemlt over K which has a unique solution - up to a unique isomorphism over K has a solution over K . )

•

Remark: When Ie

= 3 , one can suppo.se without loss of generality that (0, 1, 00). In this way, a rigid triple ( 0I , C2, 03) of rational conjugacy classes of G determines a canonical extension of K(T). Several natural questions arise in thilt\context:

(Pl, P2 , P3)

=

1 . C an one describe what hap'Rens when K is a local field? We will do this (in a special case) for K

=

R-1n §8 .4.

2. Can one describe the decomposit'ion group above �? For example, if G is the Monster, and (01 1 Oz, 03) = (aA, 3 B, 29A) as in §7. 4. 7, then the decomposition group Di above Pi must be contained in the normalizer of an element of the class 2A. This normalizer is 2 x B, where B is the "Baby Monster" sporadic group. Aside from this, nothing seems to be known about Di . Exercise: Show that the alternating group A,. (11.

any rigid family (Ch

8.2 8.2.1

. . •

,

=

4, 5, 6, 7, 8) doe. not contain

CAl) of ra.tional. conjugacy classes (use remark 1 of §7.3).

Two variants First variant

Th. 8 . 1 . 1 can be generalized to the case where the classes are only K-rat�ona1. More precisely, let US fix a choice of primitive Nth rootl of unity over It. i.e. ,

84

Chapter

8. The rigidity

method

,�rbit under Gal ( R"/ K) of primitive Nth roots of unity, T h is amounts to a K- irredu ci bl e factor of the nth cyclotomic polynomial tP"", ( For example, if K = Q( V5) , N = 5, t he cyclotomic polynomial q,s factors aa an

choosing

=

q,s(X )

X" + X3 + X'J + X + 1

=

(X

2

+ 1

1

+2v5 X + l )(X:Z + -2 v's X + 1 ).

A choi ce of 5th roots of unity is just the same as choosing a s quare root of 5 in Q( v'5).) Such a choice determines a generator :cs for each inertia group at Pi (which is well- defined up to conjuga.tion in 'irK ) ' After such a. choi ce haa been made, one has:

I;.

8.2. 1 If 01 , , Ole is a rigid family of K -TCltional classes of G, and PI ) . . OJ p.. are K-rAtiontd points of PI , then there is (I regular G-co1Jering o --+ PI de.fine.d over K 'UJhich is unramified outside {Pi t . . . , FA:} anti such. that the %s-generator of the. inertia group above. Pi belongs to the class Ct . This c01Jering is un.iqu.e.ly defined up to a unique G-isomorphism.

Theorem

• • •

The proof is e&sentiall y the same as that of

tho 8.1 . 1 .

Since conjugacy classes are always ra.tional over the maximal cy clot orm c extension Q C1c1 of Q, on e only n eeds the rigidity condition to ensure th at a group can be realized as a Galois group over QC)'d(T). This property is known for : •

Most of the classical Cheva.lley groups over finite fields (Belyi [Be2) ) i

•

All th e sporadic groupSj

•

Most of the exceptional groups t wi st ed fOrIns

8. 2.2

'lG'll J D. ,

and

G'l . F.. , ESt E7 and E. ( and also the 'l E6 ) over fini t e fields (Malle [MUD.

Second variant

The assumption that the conj ugacy cl as ses are ration al is often too restrictive for applications. The fol lowi ng variant of t h o 8. 1 . 1 is useful in practice:

8.2.2 Let (G1 , G'l l G3) be a rigid triple of conjugacy classes of 0, with GI rAtional and G2 and G3 conjugate to each. other over a quadratic fie ld Q( v'D) . Let PI E PI(Q) , P2, PJ E PI (Q( v'D», with. P2 ani Fa conjugate to ea.ch other. Then there is a regular O-eztension of Q(T) 'UJhich is ramified only a.t PI P:z and PJ I and such that the canonical generator oj the inertia group at Pi ( which i s well-defined after a choi ce of roots of unity over Q( VD») is in G, .

Theorem

�

C or ollary

8.2.3 The group 0 has prope.rty GalT .

8.3. Examples

85

The proof of tho 8.2.2 is similar to that of tho 8.1.1. The set H C Hom('lf , G) is defined in the same way. The key point is to prove that H is Itill preserved under the action of 'lfQ. •

H

a

E

1fQ is trivial on Q( ¥D), then

fixes the choice of roots of unity over before. •

a

fixes

Q( ../D).

PI . P2

and

Pl,

and also

Hence a preserves

H, as

f.1 is not trivial on Q( v'D), then u interchanges Pz and Pa, and hence lz and 13• But a alao changes the choice of roots of unitYt and these effects compensate each other.

H

Remarks: l. The assumptions

on the number of classes and the field of rationality are only put to simplify the proof, and b ecause this is the principal case which occurs in practice. In fact , the same conclusion holds in greater generality, e.g. , if { CI , . . . , C�:} is stable under the action of Ga1( Q/Q), and the map { Ph " " P.} --+ {Ci t . . . , e.. } defined by � � et. is an anti-isomorphism of Gal( Q/ Q )-sets. For other variants of tho

2.

8.3

8.1.1

using the braid groUPJ see [Fr 1) and [Mal1 .

Examples see

Here also, we only give a few examples. For more,

8.3.1

The symmetric group

[Ma3] .

S"

Sn has a rigid triple of conjuga.cy when (k, n) == 1. The covering PI --+ PI given by 1 )n-ok has ramification of this type, namely:

Recall from §7.4,1 that the symmetric group classes

X

1-+

(nA, 2A, C(I» ,

X l( X .-

{ !:� t

Hence, by tho

8.1 .1 ,

=

��)

kiCk - n )",-1I11-n

2A

the polynomial

Xl( X - I ),,-1 - T haa Galois group S" over Q(T) when (1:t n) the splitting field of the equation

Q { T 1/C J I'r).

=

1.

=

0

Note that when

(k, n)

::::

I #:

1,

Xl ( X _ 1 )"'-· - T = 0

contains Its Galois group is strictly smaller than S" (and the extension is not regular when I > 2 ) .

86

Chapter 8. The rigidity method

8.3 .2

The alternating group As

Recall from

§ 7. 4 . 2

that the triple

of conj ugacy classes in A5 is

( 2 A , 3A, 5A)

Q( v's) (but not over Q ) . By tho 8.2.1 , there is a regular extension of Q( v'5)(T) with Galois group AS J and ramification of type ( 2 A, 3A, 5 A) . The corresponding curve C has genus 0 (but is not isomorphic to Pl over Q( .J5), d. [ Se5 » . rigid . The conjugacy class SA is rational over

The action of As on

C

can be realized geometrically

Consider the variety i n P 4 defined by the equations:

{

Xl + . . . +

X5

==

u.s

follows (loc. ci t. ) .

0

X: + . . . + X; := O.

Since the first equation is linear, this variety can be viewed as a quadric hy.

The variety of lines on thi s quadric is a curve over Q which Q(.J5) to the disj oint union of two curves of genus 0 which are conjugate over Q( v's). The obvious action of 55 on V (permuting coordinates) induces an action of 55 on this curve. The extension of Q (T)

persurface in

P3.

becomes isomorphic over

corresponding to this curve is a non-regular extension with Galois group 55, which contains

Q( v'5);

Q(v'5)(T).

it can also be viewed as a regular As-extension of

An As-covering of PI with ramification (2A, 5 A , 58) can b e realized by

taking an Ss-covering with ramification of typ e

X

t-+

(2, 4, 5) (e.g . , the covering §4.5). This defines a

X5 - X· ) and using the double group trick (cr.

regular As�covering of Pl over

Q,

with two ramification points conjugate over

this is the situation of tho 8.2 .2 . This covering can b e shown to be isomorphic to the Bring curve, defined in p. by the homogeneous equations

Q ( V5);

{

Xl + . . . + Xs = 0, X; + . . . + Xi = 0, X� + . . . + X3

=

0,

cf. §4.4, exercise.

Exercise: Let C be the Bring cu.rve in p .. (see above) . a) Let E be the quotient of C by the group of order 2 generated by the transposition ( 12) in S5 - Show that E is isomorphic to the elliptic curve defined in P by the 2 homogeneouB equation; (:t3 + (put

x

:=

:t 3, y

y3 + ,z 3 ) + (:t2y + :t2,z + y2,z =

:t4.

=

,z

=

+

y2z + ,z 2 Z + .z2y) + zy.z

0,

)

Show tha.t this curve is Q�isomorphic to the curve

+

xy

:s .

50E of [ANVERS), p.86, with minimal equation

y2

=

+

y

=

X3 - X -

2

8.3. Examples

87

- 52 /2 of 55 on C product of 4 copies 01 E.

and i-invariant

.

b) Use the action

to show tha.t the

Ja.cobian of C

is Q�isogenous to the

C by A.. i s Q-isomorphic t o the elliptic c u rve 50H of i-invariant 2- 1 5 . 5 . 2113; this curve is 15.. isogenou a to E. d) Show tha.t C has good reduction (mod p) for p ¢ 2, S. If Np( C) (resp Np( E)) de�otes the number of point t on C (resp. E) modulo Pt deduce from b ) that c) Show that the quotient o f

[ANVERS],

lac. cit. , with

Np(C)

Use

[ANVERS]. p.1l7, to

=

4 Np( E) -

3 - 3".

construct a table giving

3 , I 7 1 11 1 1 1 " ' 1

Np(C) for

P<

100:

3 120 30 1 97 0 1 8 1 89 9 Check these values by determining the polynomials X 5 + aX + b over FP I wit h ( CI, b) #:- (0, 0) , which have all their roots rational over Fp' For instance, if p = 3, there is only one such polynomial. (up t o replacing « by CIt" an d b by W 5 , with 8 Np( C)

t e

F;), namely: Xi

This fits with

+

l lX + 11

0

==

24

30

.,.

(X + 33)(X + 13)(X - 4)(X - 20)(X - 22).

Np( C) 120. C has semi.. stable =

(over F. ) with the union of two co pies of PI intersecting each other at the five points of P I(F4) ; describe the action of S" on this curve, using the fact tha.t As is

e) Show that

isomorphi c to

8.3_3

reduction at

2, the reduced

curve being isomorphic

SL2(F4 ) '

The group

PSL2(Fp)

Th. 8. 2.2 applied to the rigid triples of conjug acy classes (2A, pA, pB) (when � - 1 ) and (3A, pA, pB ) ( when ( � ) = -1) .hows that there are regular PSL2(Fp)-exteDlions of Q(T) with ramification of t hi s shape. These corre­ spond to the Shih coverings with N :::: 2 and N = 3 (c!. §5. 1 ). The rigidity method do cs not predict the existence of the Shi h coverings relat ed to N == 7, which are ramified at four p oint s . A coveri ng of PI h aving ramification type (2A, 3A, pAl is gi ven by t he cov­ ering of modular curves defined over Q( H), X (p) --. XCI). T hi s covering has G alois group P SL2(Fp) over Q( #); t he rigidity property shows that it is the only PSL2(Fp )- coverin g with this ramification type. This was first

(�)

pointed out by Heeke [He] . In partiCUlar, t h e rigid t riple (2A, 3A, 7A) in covering of

functi on field E is which

is not

a.

regular.

PSL2(F7) gives rise to the Klein

Q( v:::? ) and having Galois group P SL2(F.,). Its Galo is extension of Q(T ) with Galois group PGL2 (F7),

PI defined over

Chapter

88 8.3.4

8.

The rigidity method

The GalT prop erty for the smallest simple groups

The following table lists the smallest ten non- abelia.n simple groups. All of them, except the last one SL2(F 1 S), arc known to have the GalT property. The

last

column of the table indicates why.

�

GalT property

order

group =

SL2(F.) = PSL2(F,.)

60 = 2 · 3 · 5 ==

23 3 7

SL3(F2) = PSL2( F1) As = PSL2(F9) SL:z(F8) PSL2(Fu )

360 = 23 32 5 504 ::;; 23 3 2 7 660 = 22 3 . 5 . 1 1

PSL,(F13)

1 092

=

PSL2(F 1 7)

2448

=

AT

PSL2(F 19)

2520 = 23 . 32 5 . 7 3420 = 22 3 2 · 5 · 19

S L2 (F1 8)

4080

1 68

•

.

•

•

•

•

.

22 · 3 . 7 · 13 24 a 2 1 7 •

24

. 3 5 17 .

(§§5. 1 , 8.3.3); Shih with N = 2 (§ §5. 1, 8 a 3)i Shih with N 3 (§5. 1 , 8.3.3); Hilbert (§4.5); Shih with N = 2, 3 (§§5. 1, 8 . a.a)i .

=

•

=

Hilbert (§4.5);

Th. 8.2.2 and 17.4.4; Shih wi th N = 2

.

•

•

Hilbert (§§4.5, 8.3.2) Shih with N = 2, 3 (§§5.1, 8.3.3); Shih wi t h N = 3 (§§5. 1 , 8.3.3)i

.

?

In addition, it has been shown that all the ap oradic simple groups satisfy the GalT property, with the possible exception of the Mathieu group M23 1 cf.

[Hunt] ,

8.4 8 �4. 1

[Ma3] ,

{Pal .

Local

properties

P reliminaries

?f' : C ---+ P 1 be a G-covering defined over a. field K which is complete with respect to a. real valuation. Let z be a. point of Pl(K) S, where S is the ramification locus of the covering. The fiber A;,: at :z: is an etale K algebra. wit h action of G; its structure 18 defined by a. continuous homomorphism

Let

-

.

"'� : Gal ( R / K)

which

is well·defined up

to inner

--+

G,

H the by conjugation;

conj ugation in G. Let us denote by

quotient of the set of such homomorphisms we endow H wi t h the discrete topology.

by the

action of G

8.4. Local

properties

89

Proposition 8.4.1 The map from Pl (K) to H defined by :c .... .. is contan­ uous with respect to the topology 0/ PI (K) induced by the l1cUu4tion on K . (In words, for every :c E P1 (K)t there is a neigbourhood U of :c Galois algebras A. and A. are isomorphic, for all y E U. )

other the

This

can be deduced (with some care)

P(X )

==

II(X

-

at)

from Krasner's

an d Q(X)

=

such

that

lemma: if

II(X

-

B�

as

polynomials over K, with a.a and fJi Jufficiently close for each then K(al ) == K(f31 ). See [Sal]. Another way to prove prop. 8.4. 1 is as follows: the map C ---+ Pl gives an extension O. c O., where O. is the local ring at :c, and O� i. the semi-local ring abo\re :c. Let Mz be the maximal ideal of O�. The quotient Oz/M.O. is A One has a natural inclusion O2 C O�ol, where O� denotes the ring of power series with coefficients in K which converge in a neighbourhood of :c. Since O� is Henselian (d. [Raj , p. 79), and the extension is etale, one has: are irreducible

i,

••

This means that, for its analytic st ruct ure , product. The proposition follows. By the discussion above, we of open sets:

may write

Pl (K) - S

=

where U. = {t E P1 (K) - SI;, = ;}. the universal Z/2Z-covering defined by

P1 (Qp)

-

{O, oo} int o four

non-isomorphic

Remark: naud:

covering

Pi (K) - S as

U

�H

0

PI is locally a

a finite disj oint

union

U.,

For example. if

:c

----+

t-+

:c 2

K

=

QPJ

P

�

2,

gives a decomposition of open and dosed pieces, corresponding to the four

Galois algebras of rank 2 over Qp.

A propos of the U� , let us mention an unpublished result

if the

order of G is

G-covering of

local field K.

8.4.2

the

a

of Ray­ prime to the residue ch aracteristi c t every unramified

rigi d polydisk becomes trivial after a finite extension of the

A problem on good reduct ion

Assume that 11" ; 0 � Pi li a G-covering obtained by the rigidity metho d (th. 8. 1 . 1 ) from rati on al points Ph ' " J Pic and conj ugacy class 01 , · . . , Cle • Let p be a prime number. Assume the reductions of the � are distinct i n

Pl (Fp).

90

Chapter 8. The rigidity metho d

Pro blem: A ssum e tha.t p does not divide the oTders of the elements of C, jor 1 , , k. Is it true tha.t the CUnlf C has 900d reduction at ]l?

i

=

. . .

This is true

at

least when l' does not divide the order of G, cf. [Bcl.

Exercise: Check that given in

the a.bove problem

8.3. 1 J 8.3.2, 8.3.3.

8 .4.3

has a positive answer

for t he examples

The real case

now restrict our attention to the case when 1f : Y --+ P1 comes from rigid triple of conj ugacy classes (Ol t 02 , Oa), and where K = R. By p rop 8.4. 1, to ea.ch connected component of PI(R) S there is a.ttached a unique conj ugacy class of involutions in G, corresponding to complex conj ugation. There are two cases, depending on the number of such connected components:

We a

.

-

Case 1 : We suppose tha.t the three conjugacy classes are rational OVer R.

The corresponding covering is ramified at exactly three real points Pl J P2 , P3, which divide the circle P1(R) into three connected components. Let us choose Zi E Oi, with Zl ZZZ3 = 1 and G = (Zit Zh :1;3) ' We assume G :F {I}, hence z, :F 1 for i = 1, 2, 3. Since the Oi are rational Over R, we have

=

=

The element S 3 thus defined is such that s �

-1

;1: 1 -� :1: , :;

1.

Th eorem 8.4. 2 The comple:;c conjugation attached to (I point :;c in the con­ nected co mp on ent (Ph P2 ) 0/ P I C R) between P1 and P2 is in the con.jugacy

clus of S3 '

81 and S 2 , which are defined in the same way as 8s, and correspond to the complex conj ugations attached to the components ( P2 t Ps) and (PI t P3) respect ively. This gives the relations:

Remark: An analogous statement holds for the involutions

=

Hence, if

31

and 33

are

non-trivial, the group generated by 81 J S3 and Z 2 is 2n. where n is the order of ::r:2. There is a single

a dihedral group of order

8.4.

Local properties

91

2n

(!onjuga.cy class of involutions in a dihedral group of order Hence, one has :

with n odd.

Corollary 8.4.3

If u"e order of Z2 is odd� then Sl and S3 tin! conjugate in G.

Corollary 8.4.4

II two of the three cla.sses 01 , O2 tlnd 03 h.al1e odd ezponent,

then the s, 'oS tire conjugate.

Example: The rigid triple of conjugacy classes ster

M

satisfies the hypotheses of cor.

8.4.4.

(2A, 3B, 29A)

corresponds to a unique class of involutions in M (the das s

notation).

Remark:

The case where

oS 3

=

1

2B,

in ATLAS =

occurs only in the case G

D"

and

odd. For, if "3 = 1 , then :t:l , 02:2 are of order 2, equal to .52 and .t 1 respectively. The group generated by Zl , 02:2, Z3 is thus a dihedral group of order 2n, where n is odd (since the group has no center).

( 01 , O2 , 03)

=

(2, 2, n ) , n

in the Mon­

Hence, complex conjugation

satisfying the rationality condition over 8.1.1, when applied G :# S3 J never gives totally real eztensiofU of Q.

The only dihedraJ group of this type

Q is G = S3. This

means that the rigidity method of tho

with three classes to a group

Case 2:

01 is ra.tional over R, and O2 and O;s

are

complex conjugate to each

other.

Then,

P l (R) - {Ph P2, P3} is connected, since P2 and P3

do not lie on P1 (R).

Hence there is a single conjugacy class of involutions in G corresponding to complex conjugation. Let (Z1 , Z2, Z3) in G such that:

E �(01 t O2 • 03)'

By rigidity, there is a unique involution

.t

Theorem 8.4.5 Complez conjugation belongs to the conjugacy clGS.! ol .t . Observe that i n this case

Proof of th o 8.4.2

.t

:#

and tho

1 . (Otherwise G would be cyclic. )

8.4.5

Choose a base point

on the (!onnected component (Ph Pz) in case geometric fundamental group of

X

=

01 , 02, Q3 denote the generators of

an d

P3

1)

and let

'I"

z

PI - { PI , P2, P3}, with

1r

on

P1(R),

(lying

denote as before the base point

02:.

corresponding to paths around PI ,

respectively. The complex conjugation CTx on

X

acts as

a.

Let

P2 1

symmetry

around the equator on the Riemann sphere. Hence ax a.cts on the generators ai , a2, 03 of 1r by: Case 1 : 01 1-+ ail , 02 1-+ a; l

Case 2:

02

1-+

Oi l .

a3

1-+

ai l

92

1f

Ch ap ter 8. Let

:

Y

Uy

----+

denote the complex conj ugation acting on Y. Since the G- coverin g

X is defi ne d over R, the following diag ram com m utes :

�

Y

'11'" 1 X Since

ux x

The rigidity metho d

=

crXJ

Y

1 .,..

X

x, t h e fiber Yz of Y over :r: is preserved by ay . The a.ction of ay

on Y:t commutes with the action of G.

The group G acts freely and transitively on Y:t, so a choice of 11

m ines a surjective map fJl : ""l(X; x )

----+

with that of G, the following di agram

is commutative:

""1

�

G

""l( X ; x )

�

G

(X; x ) ax 1

On the other h an d ,

we

E

G. Because the action of u

Yz deter­

com mute s

II

also have:

Hence the diagra.m below commutes:

""l(Xj x )

�

1fl(X; x )

�

ax !

G

l Inn(uy,,) G.

This proves the theorem.

Exel"Cise: Z

l

] "

Let

. ]

ZJ:

; , . '1

% ,

G be

Let

a finite group, genera.ted by elements 9 1 1 = 0 for i = I , . . ' k, and

(s ay ) R.e(Zi)

with

Z�

be the complex conjugates of .2'1 ,

point :r: < OJ let .,.. denote

{ Zi t ,

• •

�

t ZA: . Z t

• •

' . %�} .

. , " glc. Choose

points

%J: an d choose a real base the fundamental group ""1 ( P 1 ( C ) - Sj x ) . where S "

'1

:=

It is genera.ted by element s :t 1 . " " :til . :t� , . , . , :t�] where :t,

of p aths going in a straight line from :t to Zi (resp , z� ) and going around this point in the positive direction. These generators, together with the rela.tion :t I ' :tJc:t� , :::: 1 t give a presentation for .,.. . Let
(resp . :t� ) denotes the homotopy class

, •

. •

z�

8.4. Local properties 8 .4.4

93

The p-adic case: a theorem of Harbat er

Let p be

& p ri me number.

Theorem 8.4.6 ( [Harb] ) E'Very finite

e:tension of Q,,(T).

group

is �e Ga.lois

group

of a regular

proof shows more, namely tha.t for every finite group G, there exists an a.bsolutely irreducible G-covering X --+ P1 over Qp having a. "base poi nt" , i.e., an unramified point P E Pl ( Qp) which is the image of a Qp-point of X. Call R,. this property of G . The theorem clearly follows from the following two as sert ions � (i) Every cyclic group has property Rp. (ii) If G is generated by two sub groups G1 and G2 having property Jl"J then G has property Rp. Assertion (i) is easy (and true over Q, as the con struction of §4.2 shows). Assertion (ii) is proved by a gluin g process which uses rigid analytic geometry. Namely, let X. ------t P I (i = 1, 2) be a G.-covering as above with a. base point Pt . By removing a small neighborhood of .l1 one gets a rj gi d analytic Gicovering Yi ---+ U. where U, is a p-adic disk; thjs covering has the further property t hat it is trivial on Ui - U; where U; is a smaller disk contained in Ui­ Let Wt -+ Ui be the (non-connected ) G-covering of Ui defined by induction from Gt to G. (It is a d isjoi nt union of IG/Gi l copies of \1 .) One then embeds Ul and U2 as disjoint disks in P I , and defines a rigid analytic G-covering of P 1 by gluing together Wi on U1 ) W� on U2 , and t he trivial G-covering on PI U; u;. H thi s is done properly, the resulting G- covering W is absolutely irreducible. By the "GAGAu t heorem in the rigid analytic se t ting (d. [Ki] , [Ko])� this covering is algebraic, and (ii) follows.

The

-

-

Remark: Harbater's original proof uses "formal GAGAn instead of "rigid GAGAn ; the idea is the same.

Chapter 9 The for:rn Tr(x2) and its applicat ions 9.1

Preliminaries

Galois cohomology (mod 2)

9.1.1 Let

K

be a field

of characteristic I- 2, and let GK be Gal( K./ K), where K. K. We will be interested. in the Galois cohomology of

is a separable closure of

K

by

modulo 2; for brevity, let us denote the cohomology groups

Ir( GK ).

In the case where

interpretation of

Hi(GK ):

i

=

BI ( GK )

=

K-/Kd,

H2 (GK )

=

B r2 ( K ),

where B r 2 ( K) is the 2-torsion in the Brauer group of K. If a E

by (a) the corresponding element

to the

9 . 1 .2 Let

f

E

= 0,

we denote

(a. b ) i n Br2 (K)

defined by

H2(GK )

class of the quaternion algebra

i2

K· .

of Hl (GK). The cup-product

(a)( b) corresponds

Ir( GK , Z/2Z)

1 , 2, Kummer theory provides the following

?

=

b,

ij

=

-ii.

Quadratic forms

be a non-degenerate quadratic form over

K of rank

an appropriate basis, we may write

f=

"

L tl,X!,

with tli E

1=1

95

K· .

n

� 1. B y 'choosing

Chapter 9. The form Tr (:2) and its appli c ations

96

The element (1 + (a l ) ) ( 1 + (a2)) ' " (1 + (an)) in the cohomology ring H*(GK) depends only o n f . One defines the i- t h Stiefel-Whitney class Wi of f by:

In

particular I we have: Wl

=

L(ai) 'II fti) =

W2

:::::

=

(Disc(f)

2: (a,)(a;). i.
The cohomology class W 2 is known as the Hasse (or Witt) invariant of the quadratic form f. If K is a number field, then f is completely chara.cterized by ita rank, signature, and the invari an t s WI and W2' (The same is true for ar bitrary K, when n $ 3.) The following results can be found in [Sch] , pp . 211-216.

(Springer) If two quadratic forms over K become equivalent over an odd-degffe =tension K' of K, then they are already equivalent over K.

Theorem 9. 1.1

(For

a generaliza.tion

to hermitian forms, see (BL] . )

Let us now consider K(T), where T is an indeterminate. If v i s a pla.ce of K(T) which is trivial on K and ¥ 00, there is a unique uniformizing par ameter '1'" at 'fJ which is monic and irreducible in K[T] . Let K(v) = K[T] /(w,,) den ot e the residue field a.t Vt and let 4 1-+ a be the reduction map K[T] � K (v). If ! � E�l a.Xl is a quadrat i c form over K(T), we may assume (since the ai can be modified by squares) tha.t v ( a,:) = 0 or 1. The K(v)-quadratic forms:

a1 ( !) ;::::

L

"(0.)=0

a;,

X:,

02(/) =

L (a,/'If,,) X:

.(Cli )=l

are called the first and second residues of f. One shows ( cf . e .g. [S ch}, p . 209) tha.t their images in the Witt group W (K ( v)) do not depend on the chosen representation of f as E a,Xl - (Recall that the Witt group WeLl of a field L is the Grothendieck group of the set of qu a.d rati c forms over L, with the hyperbolic forms identified to 0. ) The orem 9. 1.2 ( Milnor) If a quadratic form l over K(T) has second residue o Ilt all plilces oj K(T ) e:cept 00, then 1 is equivalent to 4 qUlldratic form over

K.

/

9.1.

Preliminaries

97

(More precisely, one has an exact o

--+

W CK)

---t

A qua.dratic form over a

sequence:

W( K(T))

ring

discriminant is invert ible in R.

R is

Theorem 9 . 1 . 3 (Harder) A strictly

ri""9 K [T) comes from K.

This over

theorem can be formulated. more

--+

said

U W( K (v)

v�oo

---+

0 .)

t o be strictly non-degenerate if it s

non-degenerAte quadratic form over the suggestively:

a quadratic vector bundle

Al is conatant. (For a generalization to ot her type! of bundles, see

9.1.3

We need

[RRJ . )

Cohomology of Sn only

H'(Sn) for i

=

1 , 2. These

groups

are

well·known:

Z/2Z, n � 2, Z/2 ED Z/2Z, n � 4 The non- t rivial

(Schur) .

H1 ( S,, ) is the signature homomorphism

element in

The cohomology group H2($,, ), n � 4, has a Z/2Z-basis given by � (cup-product) and an element 3n corresponding to the central extension

1

......-..

O2

--+

5" --+ S" ----+

which is charact erized by the propert ies : 1 . A transposition in 8ft lift s to an element of order

2. A pro duct of two disjoint

transpositions lifts to

an

.

Et&

1

2 in Sn . elem ent of order 4 in 5",.

(The element e,.. . � correspond s to the extension s:. of Sn. ob t ai ned by taking the pullback: 04 s'" !

$"

!

{±l }.

Thia extension is characterized by t he property that a transposition lifts to an elem en t of order 4, while a p rodu ct of two disjoint transpositions lifts to an element of order 2.) The im age of Eon by the restriction map Hl (Sn) ---. Hl (An) i s zero. The cohomology group H2 (A,,) is isomorphic to Z/2Z for n � 4 and is generat ed

Chapter 9. The form

98 4n

by

( or

Tr (:z:2) and its

applications

Res ( " n) . The corresponding central extension of An is denoted by Aft 2 . � in ATLAS' style); it is a subgroup of index 2 of Sn '

=

by

The

En and Sn can be gi ven the following topological ---. On(R) gives a map of classifyi ng spaces

co homology classes

interpret ations:

the ma.p

and corres ponding

wh ere Wi

Sn

maps on

the

cohomology rings.

is the ith Stiefel-Whitney

element En

Exercises:

in B l (BS".)

=

class.

HI (Sn )� and

W:z

But

The class gives Sn .

WI corresponds to

the

SL:z(F3), As � SL2(FIi ), and As � SL:z(Fg). of Sft by {±1} corresponding to the element 8ft + 4a 4a. of H2(Sn ). ( a ) Show that 5;. � GL 2 (F3). (b) Show tha.t an y outer automorphism o f Se lifts to an isomorphism of 56 onto 1. Show that

2. Let Sn be

A4

::::

the

centra.l extenaion

•

56.

(c)

Show that the groups

9 .2

Sn and Sn are not

isomorphic if n � 4, n ':f:. 6.

The quadrat ic form Tr

(x2)

an etale K-algebra of finite rank

n over Kj it is a. prod uc t of sepa.­ of K. There is a dictionary b etween such algebras and conjugacy classes of homo mor p hi sm s e : GK ---t Sft, which works as fol lows : given E, let �(E) be the set of K-algebra homomorphisms E � K The set tI>( E) is of cardinality "" and t he natural a.ction of GK on ¢(E) gives the desired homo m orphism e : GK --+ Sn , after i dent ifying �(E) with {l" , . , n}. C onvers ely . E can be co nstructed as the twist of the split algebra K x . . . x K n by the l-cocycle e : GK --+ Sn = Aut ( K ). The i m age of e is the G alois group of the smallest extension of K over which the algebra E splits; if E is a. field, it is Gal( E�eJ. / K).

Let

E

be

rable field extensions

•.

The func ti on 3: � Tr (x 2 ) defines a non degener at e quadratic form QE of rank n over K. This invariant was st u d ied extensively by 19th century ma.th­ ematici ans such as Jacobi and Hermite. �

Theorem 9.2.1 Let E/ K ( T) be a. finite s epa.rcble extension of degree

n.

L�t

G c Sn be the COfTesponding Ga.lois group . As.'J1,me tha.t, for a.ll places tI of K ( T ) not equa.l to 00 , the order of the inertia. group at v is odd. Then QE is consta.nt,

i. e.,

comes from

(I

form over

K.

9.3.

Application to extensions with Galois group k

99

Let A be the in tegral closure of K[T] in E, and denote by 'D the different. Using the fact that the inertia groups are of odd order, one can show that V- 1 is the square of a fractional ideal,

One checks that 0 B induces a strictly Don-degenerate quadratic form over KIT ] on the K{T)-module A. By tho 9 . 1 .3 the result follows. ( O ne may also show that the second residues of Q B are 0, and app ly Milnor's theorem 9 . 1 .2.)

a regular Galois extension of Q(T) with Ga.lois group the Monster, obtained from the rigid family (2A, 3B, 29A), cf. §7.4.7. Since two of the conjugacy classes in this family have odd exponent, the form QE comes from Q , and does not depend on T. One can prove that it is hyperbolic.

Example: Let E be

The fol lowin g theorem 18 proved in [Se6] .

Theorem 9.2.2

tlSsociated to

a

form oj E . Then : 1.

1lJl ( Q S) :; W2( QS) =

!.

Let E be tln ittll e K -algebra of ranJc n tlnd discrimintlnt d e : GK ---+ SnJ an.d let Os denote the trace

homomorphism

e · En., e · sft + (2)(d)

Suppose that G = e ( GK ) is contained in A ' B i.e. , that d is a square. By tho 9.2.2, we have tVl (QB) = 0 and w,(QE') ;;; e· o.,.. The element e·o.,. is the obstruction to lifting the homomorphism G K ----+ A n to a homomorphism

GK

--.

An. Hence, we have

Corollary 9.2.3

GK

----+

9.3

Aft if and

The homomorphism e � GK only if

the

Witt

----t An lifta to a homomorphism invariant 1.U3 ( Q B ) is O.

Applicat ion t o extensions with Galois group

An

The previous corollary applies when e is surjective: an extension E of K of degree n with Galois group A.,. can be embedded in an A,,-extension if and only if W2(Q S) ::; O. This will be used to show the following result: Theorem 9 . 3. 1 (Mestre, {Me2] )

oJl

n.

The group

.4"

hD.S

the GalT-property for

Chapter 9. The

100

( This was already shown for

n ==

0,

form

Tr (:r 2 ) and its applica.ti on s

1 (mod 8), a.nd for some other

n,

by Vila

[Vi ] . ) The pr oof constructs a covering C ---+ P I of degree n w hose G alois closure has Galois group A,.. , and which has the following addi ti on al properties :

1.

PI

There is a point of

whose inverse i m age is a set

t he a.; a.re rational and distinct.

2. The non-trivial inertia. groups a.re all Let E 1 K(T) be the

degree n

{al "

.

. , 4n h

gener ate d by cycles of order

extension corresponding

to this

where

3.

covering. Con­

2 i mpli es that the quadratic form QE comes from Q , by tho 9.2. 1 . But by cODdition 1, t her e is on e rationa.l point where this quadratic form is equiv� dition

alent to the standa.rd one, L Xl . It is easy to see that this implies that Q E is

L Xl over Q(T), and hence has trivial Wit t invariant j by can thus solve the embedding problem for this extension.

equivalent to 9.2.3, we

a.

Let us say that a. property E of

p

=

n x +

is generally true if there exists P has the

od d)

prop e rty E

for all

relies on the foll owing:

a

polynom ial

SlX"- 1

+

...

+ 8n

Zariski open dense subset U

(S 1 1

.

•

cor.

•

, Sn)

of An

such that

in U. Mestre's construction ( for

n

Prop osition 9.3. 2 It is generally true th4t there e:rist polynomials Q and R in Q [T ] of degree n - 1 with the following properties :

a) Q'p - P ' Q = R2 ( i . e . , ( Q I P)' = ( RI P)2 ) . b ) P, Q, R 4re p4irwise rel4ti1Jely prime.

The zero3 b1 , , bn-1 of R in Q are distinct. d) The 1J4lues t, of Q / P at the Oi are distinct.

c)

. . •

Skdch

for

i

==

of proof: The matrix M

j is a skew

with ij - e nt ry

symmetric m atr ix

determinant and there exis t s a nOD-zerO have:

t.. � = o

j

l fli - fl,

#1.

( C. , . . . J en) in for all

Now, let Q , R be the polynomials such tha.t :

RI P

ft

=

1 /(4.. - 4;)

of odd dimension.

Ci E x-:::'

i= J

t:t.

I.

for

i =I- j

and 0

Hence it has zero

the kernel of M. We

9.3.

Application to extensions with Galoi. gTOUp A,.

One

has: =

= =

Moreover, one can

"

'�1 (X

CiC "

a,) (� - aj) t _...; c'f� ;.4; )2 + E (X -

101

"(_ 1

_ "C'_ " _Ci

i;f:; CI.i

i=l

A;

_

X - a.

_

1_ )

_

X-

a;

(Q/P),

check ( by looking at a well-chosen example) that it is gen­ rank n - 1 (so that the Ci are essentially unique) and

erally true that M haa

C), d ) hold. If (P, Q, R) are chosen as in prop. 9.3.2, with P = n(X 4i being dis ti nct , then the map PI -+ P1 given by X .......

that b) ,

-ail with

haa degree n., and is ramified at

the being gen erated by 3- cy cles . Let G c

zeros

bi

of

T

=

4i E Q, the

P(X)/Q(X) R, the ramifica.tion groups

8ft (resp. (J c S,,) be the Galoia group

Galois extension of Q(T) (resp. of Q(T)). The group G by 3. cyclea (cf. prop. 4.4.6). By lemma 4.4.4, it is equal to A..a . Hence G is equal to An or Sft. However, we have seen that the Tr (:z: 2 ) form attached to the extension is the standard form E Xl. In particular, its discriminant is a square. This shows that G is contained in Aft , hen ce G = G = Aftt QED. There is a. similar, but more complicated , construction when n is even. Exercise 2 below proves that L haa the GalT property for n even by reducing to the cue of odd n.. of the correBponding

is transitive and generated

Remarks:

1. One may also prove Mestre's theorem by showing that the Witt i nvari ant W 2 E Br2(Q(T» of the trace form haa " no polesu (because all the ramification is odd ) , and hence is constant, i.e. , belongs to Br2 (Q). Since it is 0 at the bue point, it is zero. Another possibility is to prove by a direct con stru ctio n that the trace form of E / K(T) is con st ant , cf. exerc. 1. 2. For explicit formulas related t o the above constructions , s ee [Or) , {Schn] . 3. For n. = 6, 7, the Sch ur multiplier of � is cyclic of order 6. The correspond­ ing groups 6 · As and 6 . AT also have property GalT. Thi s haa been proved by J-F. Mestre (unpublished). Exercises : 1.

Assume P, tJ, R are aa in prop.

9.3.2

(n odd).

Put

E :::: X(X ) and T :::: P(X)/Q(X}. so

that [E ; K(T)] :::: n..

a) Let A be the integral closure of K[T) in E. Show that A = K[X, l/Q{X)] and

that the different of A over

K[T] is the principal ideal generated by R(X ) 2 .

Chapter 9. The form

1 02 b) Let be the trace map. If

f

that:

E

Tr : E

K[X],

sh ow

degT Tr (Je X )/R( X ):l)

�

and its application6

K(T)

----+

that

Tr ( z:l )

Tr (f(X)/ R(X) 2)

5Up( O, deg(f)

-

belongs to K[TJ a.nd

271. + 2).

(J(X)/ R(X) 2 ) belongs to K if deg( f) :5 271. - 2. c) Let V be the n-dimenlional K-sublpa.ce of E spanned by

In particular, Tr

i/R(X ) , X/ R(X), . . . , x

n l - / R(X) .

One hu E = V ®K K(X). Show, using b), that if lIl t V, E V, then Tr ( 1II 1I2 ) E K. Conclude that the trace form of E / K (T) comea from K. 2.

Let

n

be even, let

f

:

construction above , a.nd let

and 9

:

CJ

......

PI

CJ

--.

PI

be

an

n

+ i- covering given by Meltre's

be it Galois closure. One haa maps

PI is a regular An� covering.

Show tha.t tml covering lifts to a regular

An- covering of Pl (hence An also haa property GalT ) .

Chapter 1 0 App endix: the large sieve inequality Statement of the t heorem

10.1

Let N be an integer � 1, and, for each prime p, let lip be a real number with o < lip � 1. Let A be a sub set of A = Z-, such that for all primes p,

where

(:1:1,

• •

•

Ap C AI ph denotes the reduction of A mod p. Given a vector z := , :z:,,) E Rft and N E R, we denote by A( x, N) the set of points in A t

which are contained in the cube of side length N centered at

:1: ,

i.e.,

Then:

Theorem 10. 1 . 1 (Large sieve inequality) For every D � 1/ we h.a.lIe

Taking

D

=

Nt :

Corollary 10. 1 . 2 IA(z, N}1 'S

(2N)ft/L(Nt ).

103

104

Chapter 10. Appendix:the la.rge sieve

inequ ality

Examples! 1. H vp = � for eve ry P, then

Hence IA(z, N)I « 2.

Assume there is

p E S, with 0

primes .s D:

<

C

<

(

=

L(D)

L 1) tl ���free

.....

6 - D. 7r 3

,N"- L This is a typical "large sieve" si tuation. a

set S

1.

of primes of density >

0,

such that

V" =

C for

O ne may estimate L(D) from below by summing

L(D) � 1

+

L


1

.

- v" v"

::2>

Over

�. lo g D

Hence j A(x, N)I <:: ,N"- l log N. A more careful estimate of LCD) by summing over all square--free d � D allows one to r eplace the factor log N by (lo g N )'l, with 'Y < 1 , under a mild extra condition on S, d. [Seg} , chap. 13.

3. Suppose n = 1 , and v" == 1 - ; . Then one can show that L(D) ;:::" log D, and hence IA(z, N) I <: lo:N I a weak form of the prime number theorem: howe ver, the method also allows one to conclude that in any interval of len gth N I there are at most O( lo:N ) primes . (More precisely, their number is � 2N/ lo g N, cf. [MoV] . ) Historically, a weaker form of the sjeve inequality was discovered first, where the sum giving L(D) was taken over the primes � D; this only gave interesting results in lar g e sieve situations (hence the name "lar g e sieve inequality" ). The poss ibility of using square�free d's was pointed out by Mont gomery, (Mol]. Exercise:

that p + 2

Use t h o 1 0 . 1 . 1 t o show that t h e number of "twin primes" (primes p such also prime) � N is sympt oticaJ1y < (iosNN )f ' Conclude that a

is

1

L

twin

"

prune

-

p

< 00 .

P ro of of tho 10. 1 . 1 : preliminaries

Let us assume without loss of generality that A = A(z, N). a = ( al , . . . I �), t = (tit . . . , tn ) belonging to Rn , put

Xm(t)

=

exp

G iven

vectors

(27rit ) ,= 1

a;t; .

Rn /zn by We identify A = Z N wi th the character group of the torus T a 1-+ Xa l and associate to A ACz, N) the function t/J whose Fourier expansion =

=

105

10.2. A lemma. on finite groups is:

; = LA Xca. Cle

d ition on the reduction of A mod p and the fact that A is contained in a cub e of side length N give rise to inequalities satisfied by ¢; combining these will give the sieve inequality. The

co n

A lemma on finit e groups

10.2

Let C. for' l

:5 i � It. be finite

abelian group.

=

(written a.dditively), OJ

their

character groups, t/J a function on C n Ci . Suppose there are subseh 0,: of Ci with Ind :S lIi l C, I, such that the Fourier coefficient of t; rela.tive to the character X = (X. ) E t == n 6; is 0 outside n (li . Let us call :l: E C primitive if its image

in each C,

is =f. O. Then:

Lemma 10.2.1 We have :

11:

�

primitive

1;("'1 1" � 1;(0) 1"

IJ C : V;) . .

We give the proof in the case of a single group C: the general case follows by induction on the number of facton. Write tI> = E Cx;X, the sum bei ng taken over all characters X E n. Then:

Applying the

Cauchy-Schwarz ine qual i ty, we ItI>(OW'

and hence

=

get

I E ex ' 11 2 s. E l ex l2 E 1 ,

1q,(O) 12 �

"eO

IIl CE ItI>( :t ) 12 + 1 <1>(0)12 ). z:#o

The lemma. follow s by rearranging terms in this inequality.

10.3

The D avenp ort-Halberstam t heorem

Define a distance on Rn by {:l:1 = sup l :t. l i thi s defines a distance o n the torus T = Rn /zn , which we aho denote by I I_ Let 6 > 0; a set of points {til in T

is

called

5-spa.ced if [ t, - ti l � 6' for all i =f:. i.

Chapter 10. Appendix:the

106

la.rge sieve inequality

Theorem 10.3. 1 (Davenport-Halberstam) Let rP = L C�Xl be a continuous func tion on T who.!e Fourier coefficients Cl vAn ish when .\ is outside .!ome cube E of .!ize N. Le t � E T be 6-spaced points for .!ome 6 > O. Then

�, I rP( ti)12 � 2ft sup(N, i t l i rP l l�, where 1 1 ';1 1 2 is th e L2-n o rm of rP . If 6 > 1 /2, there is at most one t, and the inequality follows from the Cauchy­

Schwa.rz i neq uality applied to the Fourier expansion of

�.

6�

that

One constructs

an

auxil iary function (J on

rP .

Let

us

Rn t such

1 . (J is continuous a.nd vanishes outside the cube 1%/ to view (J as a function on T.

<

��.

now suppose

tha.t

Th i s allows us

2. The Fourier transform of (J has a.bsolute value ;:: 1 on the cube

3. 1 1 91 1� � 2 n M'\ where M Let ..\

E

Rn

=

sup ( N,

E.

� ).

be the center of the cube E. Then one checks , by an elementary

compu ta.tion, tha.t the function 8 defined by

elsewhere. has the required properties . coefficient of

We are:

rP i

defi ne

For each ..\ E A, let

simila.rly cl ( B ) . We

ma.y thus define a continuous

Since Cl( '; )

B*9

of

B

=

and

fun ction

cl ( B )Cl(g) for every .\ g. Therefore:

rP (t,) where B, is the s et of

=

E A,

Cl( '; )

be the .\.th Fourier

have:

9 on T whose Fourier coefficients

if ..\

E

if ).

�E

E

rP is equal to the convolution product

( B ( t, - t )g ( t ) dt = r B( ti - t )g(t) dt, JT lB,

t such that It-tal

<

�.

By the Cauchy-Schwarz inequa.lity;

10.4. Combining the information t he t,

Since

are 6�lIpa.ced,

the

107

B, are disjoint. Summing over i

then

gives

E 1 t/>(t.;) 12 � 2" M" l Ig l l� � 2" M" 1 It/>1 I�, i

becaus e

I l g lI; = This

1:

co

mpletes the proof.

Remark:

In the

(Selberg, see

n.

1 0 .4

case n

=

1, the

e.g. [M02] ); it

I �:�:�r

�

"","�.

can be impraved to N + i similar improvement holds for any

factor 2 sup(N, � )

is likely that

a.

Comb ining the information

Let D be given; the set {ttl of all d-division points of T, where d ranges over positive square-free integers :-:; D, is 6-spaced, for 6 = 11 D2. Applying tho 10.3.1 to t/> ::: E"E A X,,, we have

E 1t/>(tiW' 5 2" sup( N, D2 )" I A I · •

On the

splits

other hand, for ea.ch d �

as

D square-free, the kernel T(d) of d : T

T [dJ

=

( 1 0.1) ----+

T

TI T[Pl Jlld

and its character group is AIdA ::: npl.r AlpA. Hypothesis (2) on A allows us to apply lemma 1 0.2. 1 to the restriction of t/> to T{d) . We thus obtain

Hence,

by summing over all square-free d � D,

L 1 ,p(t.) 12 i

Combining equations 10. 1

and

we

;;:: IAI2 L(D).

obtain:

(10 .2)

10.2 and cancelling a factor of IAI on both sides I AI = 0 does not pose any problem.)

gives the large sieve inequality. (The case

QED.

Remark: A similar sta.tement holds for a number field K; A is replaced by

OK x · · · X OK, where OK

denotes the ring of integers of K; the corresponding

108

Chapter 10. Appendix:the large sieve inequ ality

torus

T is

then equipped with a natural a.ction of OK . The technique of the

proof is essentia.lly the same as in the case

Exercises:

Pi (i E J)

1.

Let

A

m od

be integers � 1 such tha.t

K

=

(Pi,P, )

of zn contained in a cube of side length N. Let

Q,

=

IIi

see

[Se91 , ch. 1 2.

1 if i � j. Let A be

a

subset

be BUch that the reduction of

Pi has at most ViYi' elements. Show (by the same method as for tho 10. 1 . 1 )

that

with

L{D) ;; E II{l - Vi )ll/i, J iEJ

where the sum runs through a.ll subsets J of 1 such that

ITiE J Pi :$

D. (This ap plies

for instance when the Pi'S are the squares or the cubes of the prime numbers.) 2. Let H be the set of pairs (:2:, y) of integers 1= 0 BUch that the Hilbert symbol (z,

y)

is trivial (Le. , the conic Z2

-

zX 2

-

yy 2

=

(by using exerc . 1 . ) that the number of points of <<

N 2 / log N .

0 has

H

in

a

a.

r ational point ).

(Whether or not this bound is sharp is not known. )

For generaliz ations , see [Se lO] .

Show

cube of side length N

is

B ibliography [Abl]

[Ab21

S. Abhyankar. Coverings 0/ algebraic cu"es, Amer. J. of Math. 79 ( 1957), 825-856. S. Abhyankar . Galois theory on the line in non.zero characteristic,

Bull. A.M.S . , to appear.

[ANVERSl [ATLAS)

B.l. Birch and W. Kuyk (edit .). Modular Functions of One a.ble IV, Leet. Notes in

Ma.th. 476, Springer-Verlag, 1975.

Vari­

J.R. Conway, R.T. Curtis, S.P. Nort on , R.A. Parker, and R.A. Wilson. Atla.s of finite groups: maximal subgroups and ordinary characters for simple groups. New York: Clarendon

press,

1985.

[Be]

S. Beckmann. Ramified prime.! in the field 0/ moduli 0/ bra.nched coverings of C'U",es, l. of Algebra 125 ( 1989), 236-255.

{Bel]

G.V. Belyi. On Galois eztensioTU of a muimal cyclotomic field, Izv. Akad. Nauk SSSR 43 (1979) , 267-276 ( = Ma.th. USSR Izv. 14 ( 1 980), 247-256).

IBe2]

[BL1

G.V.

Belyi . On eztension.s of the ma:tim41 cyclotomic field having

a given classical Galois group.

J.

Crelle 341 (1983), 147-156.

E.

B ayer-Fluckiger and H.W. Lenstra. Forms in odd def}'ne eden.­ .lions and self-dual normal bases, Amer. J. of Ma.th. 1 12 ( 1990),

359-373. [Bol

E.

Bombieri . On ezponential

sums

in finite fields II, Invent. Ma.th.

47 ( 1978) , 29·39. [Bor]

A. Borel. Introduction awe groupes mann. 1 969.

[Coh ]

S.D. Cohen. The

ducib,Zity

arithmetiques.

distribution 0/ Galois groups

Paris , Her­

and Hilbert', iJTe·

theorem, Proc. London Math. Soc.(3) 43 ( 1981 ), 227-

250.

109

110

Bibliography

[CT]

J-L. Colliot-TheIene. Letter to T. Ekedahl, 9/21 /1988 , unp ub­ lished.

[CTSll

J-L. Colliot-ThelEme a.nd J-J. Sansuc. Principal homogeneo1J.S 'po.cel o.nd jfo.sque tori: tlpplico.tiOfU, J. of Algebra 106 (198 7), 148-205.

[CTS21

J-L. Colliot-Thelene and J-J. Sansuc. LtI. de!cente !ur Ie! vo.rietls rtl.tionnelle" II, Duke Math. J. 54 (1987), 375-492.

[Cr]

T. Crespo. Ezplicit const"'ction of L-type fielcUt J. of Algebra 127 (1989), 452-461 .

[CR]

C.W. Curtis and I. Reiner. Representation Theory of Finite Groups and Associative Algebras, Inters. Publ., 1962.

(Da]

H. Davenport. Multiplicative Number Theory (revised by H.L. Montgomery), Springer-Verlag, 1980.

[DD1

R. and A.Douady. Algebre et theories galoisiennesj 2/ theories galoisiennes, CEDIC /F.Natha.n, Paris, 1979.

[Del]

P. Deligne. Le su.pport du co,'mctere d 'une representtl.tion .!uper­ cuspidale, C.R. Acad. Sci. Paris 283 (1 976), 155-157.

[De2]

P. Deligne. Le grou.pe /ondo.menttl.l du complement d 'une courbe plane ntayant que des points double.! ordinaires est abeJien, Sem. B ourbaki 1979/80, no. 543.

[Ek]

T. Ekedahl. An effective 'Uersion 0/ Hilbert '.! irreducibility theo­ rem, preprint, Stockholm, 1987. (d. sem. Theorie des Nombres , Paris 1988- 1989J Birkhius er, 1990, 241-248).

[F)

R.Fricke. Lehrbuch der Algebra. III. Algebraische Zahlen, Braun­ schweig, 1928.

[Fo]

O. Fonter. Lectures on Riemann surfaces, Springer-Verlag) 1981.

[Fr1}

M.D. Fried. Fields of definition 0/ function fieltbJ a:",d Hurwitz fo.milie� - Groupl as Galois group.!, Comm. Alg. 5 (1977), 17-82.

[Fr21

M. D. Fried. Rigidity and o.pplications 0/ the classifico.tion of sim-­ pIe group.! to monodromy, preprint.

[Ga]

P. X. Gallagher. The ltl.rge sieve and probabilistic Galois theory, Proe. Symp. Pure Math. 24, 91-101 , A.M.S., 1973.

Bibliography

111

[G H]

P. Griffiths and J. Harri s . P rinciples of Algebraic Geometry, John Wiley and Sons, New York, 1978.

[GRJ

H. Grauert and R. Remmert. ( 1958 ), 245-3 18.

[Hub]

D.

Harba.ter.

Komple2:e R iume,

Math. Ann. 136

G6lois covering' of the arithmetic line, Lect.

in Ma.th. 1240, Springer�Verlag, 1987, 1 65- 195.

[Hal

R. Hartshorne. Algebraic Geometry, Springer-Verlag, 1977.

[He]

E. Heeke.

Die eindeutige Be,timmung der Mod"UlfunJctionen q-ter Stufe dureh. algebraische Eigenschajten, Mat h . Ann. 1 1 1 ( 1935),

293-301 (Hi]

Notes

(=

Math. Werke, 568-576).

D. Hilbert . Ueber die Irreduzihilittit gcnzer rationaler JfUnA:tionen mit ganzzahligen Koeffizienten, J. Crelle 110 ( 1892) . 104- 129 ( =

Ges. Abh. II, 264-286). [Ho}

G. Hoyden-Siedersleben. Realisierung der Janlcogruppen J1 unci J2 ala Galoisgru.ppen Uber Q, J. of Algebra 97 ( 1 985), 14-22.

[Hu1

B. Huppert . Endliche Gruppen I, Springer-Verlag, 1967.

[Hunt1

D . C. Hunt. Rational rigidity and the ,poradic groUp', J. of Algebra. 99 ( 198 6), 577-592.

[ILK1

V.V. Ishanov, B.B. Lur'e, D.K. Fa.ddeev. Embedding problems in Galois theory, Moscow, 1990.

[Is 1

V. V. !shanov.

tent 1cerne�

On the ,emi-direct embedding problem with. nilpo­

Izv . Akad. Na.uk SSSR 40 ( 1976), 3-25, ( ::M a.t h .

USSR Izv. 10 ( 1 976),1-23).

[J 1 1

C. Jordan. Traite des substitutions et des equations algebriques, Gauthiers- Villars� PariSj 1870; 2eme edition, Paris, 1957.

[J2)

C. Jordan. Recherche... .!ur le' Jub.ditution.s, J. Liouville 1 7 ( 18 72), 351-367 ( = Oeuvres, I, no. 52) .

[J3} [Jou)

C. Jordan.

Sur la limite de tren.sitiviU des groupe.. nan altcrnl"

Bull. Soc. Math. France 1 ( 1 873) , 40-71

(=

Oeuvres , I, no. 57)

J-P. Jouanolou. Theoremes de Bertini et a.pplications, Birkhiuser, 1983.

Bibliography

1 12 [JY]

C. U. Jensen and N. Yui. QU4ternion e:r:tensions, Algebraic Geom­ etry and Commutative Algebra, in honor of M. Nagata ( 1987). 155-1 82.

(K i]

R. Kiehl. Der EndliehJceiua.tz fUr eigentliche Abbildungen i n der nichtarchimetlischen Fu.n1ctionentheorie,

Inv. Math. 2 ( 1 967 ) , 191-

214. [Ko]

U.

Kopf. Uber

eigentliche Familien algebraischer Varietiiten uber

affinoitlen Riiumen,

Sch.riftenreihe Univ. Munster, 2. serie Heft

1 974. [KN]

W.

Krull and J. Neukirch.

Die Strulctur tler 4bsoluten Galois­

gruppe iiber dem Kiirper R{t),

Kuyk. 1 12- 1 24.

7,

M ath. Ann. 193 (1971), 1 97-209. J. of Algebra 14 ( 1 970),

[Ku]

W.

[L]

S. Lang. Fundamentals of Diophantine Geometry, Springer-Ver­ lag, New York, 1 983.

[Le]

H. W.

Eztensions de corps hilberuens,

Lenstra. Rational functions invariant Invent. Math. 25 (1974), 299-325.

under a finite. abe.lia.n

group,

[LM] [Mae]

M.E. LaMacchia. Polynomials

mun. Alg. a (1980), 983-992.

with Galois group PSL( 2, 7 ), Com�

T. Maeda. Noether 's problem for As ,

J. of Algebra 1 25 ( 1 989),

41 8-430. Maha.t. Zwei Aspekte gebra. 96 ( 1985), 499-531 .

[Mall

B.H.

[M a2]

B.H.

[Ma3]

B.H.

[Ma4]

B.H.

[Me11 [Me21

Mabat.

1construlctiver Galoutheorie, J. of

Al­

Topologische Automorphumen in der ko nstT'ZLktiver

Ga.loistheoriet

J. Crelle 371 (1986), 16-45.

Matzat . Konstruktive Galoistheorie, Led. Notes in 1 284, S pringer-Verlag, 1987.

Math.

Matzat. Konstru.lction von Zahlkorpem mit der Galougruppe Ml1 uber Q( yCIT), Man. Math. 27 (1 979), 103- 1 1 1 .

J -F. Mestre. Co urbes hyperelliptiques d C . R.Acad.Sci. Paris 307 ( 1988), 721-724. J-F. Mestre.

Eztensions regulieres de

An , J. of Algebra 131 ( 1 990), 483-495.

multiplications reelles,

Q(T)

de gro upe. de Galois

Bibliogmphy

1 13

[MIl 1

G. Malle. Eueptional grouP! Crelle 392 (1988) , 70-109.

[MI2]

G . Malle. Genus e%tensions, Man.

[Mo11

a.s

Galois

group�, J.

zero tra:nsl4tel of three point ramified

Galois

Math. 71 ( 1991 ), 97.,,1 11 .

H.L. Montgomery. A note on the large .neve, J. London Soc.

[Mo2)

of Lie type

Mat h.

43 (1968). 93-98.

H.L. Montgomery. The analy tic principle of the large liel1e, Bull .

A.M.S. 84 (1978), 547-567 .

[MoV]

H.L. Mongomery and R.C . Vaughan. The large �ie1Je, Mathemat­

ica 20 (1973), 119-134.

[Ne1 [Noe1

On �oluable number fields.

E. Noether.

Gleich:u.ngen mit 110rge!chriebener

Ann.

[De} [ O s] [Pal

[Prj

Invent. Math. 53 ( 1979),

J. Neukirch. 135-164.

Gruppe , Math .

78 ( 1918), 221-229 (=Ges. Abh. no. (2).

J. Oesterle.

Nombn:! de TafTUlgD:wa et groupel unipotent" en

a.cteristique p. Invent. Math

H. D.ada. The Galo is

car­

78 (1984), 13-88.

grouP! of the polynomials zn + az' +

b, J.

Number Theory 25 (1987 ), 230-238. H. Pahlings. Some Ipora4ic group" as Mat. Univ.

Galois group�, Rend. Sem. Padova 79 ( 1988), 97- 107; II, ibid. 82 (1989), 163-171;

correction,

to appear.

B. Pryzwara. Die Operation der Huf"Witzschen Zopfgruppe auf den Erzeugerniensyltemlclaslen endlicher Gruppen,

Diplomarbeit,

Karlsruhe, 1988. [Ra] [Re1

M. Raynaud. Anneaux locaux henseliens . Leet.Notes 169, Springer-Verlag, 1970.

in

Math.

H. Reichardt. Konstruktion von ZtJAlkorpern mit gegebener Ga­ loisgruppe von Primzo.hil'otenzordnung, J. Crelle 1 77 (1937), 1-5.

[RR]

M. S . Ragbunathan and A. Ramanathan. PrincipaJ bundlel on the a.ffine line, Proc . Indian Acad. Sci. ( Math . Sci.) 93 ( 1984), 137· 145.

[Sal]

D. Saltman. Generic Galois eztensions and problenu ory, A dv. Math. 43 ( 1982), 250-283.

in field the...

1 14

Bibliography

(Sa2]

D. Salt man. Naetner 'J p ro blem over an algebraically cloJed fie.ld, Invent . Ma.th. 77 (1984), 71-84.

[Sch]

W. Scharlau. Quadratic and Hermitian Forms, Springer-Verla.g, 1985.

[Schm]

W.M. Schmidt. Integer points on hyper.Jurface.J, Monat . fiir Math. 102 (1986), pp. 27-58 .

[Schn}

L. Schneps . Ezplicit construction of eztensions of K ( t) of Galois group A,. for n odd, J. of Algebra, to appear.

[Sel1

J·P. Serre. Cohomologie Galoisienne, Leet. Notes in Math. Springer-Verlag, 1 973 ( 4th edition).

[Se2]

J·P. Serre. Local Fields, GTM 67, Springer� Verlag, 1979.

[S e3]

J-P. Serre. Algebraic Groups and Class Fields, GTM 1 1 7, Spring. er-Verlag. 1988.

[ Se4]

J-P. Serre. Geametrie algebrique et g eo metrie analytique, Ann. Inst. Fourier 6 (1 956)1 1-42 (=C.P. no. 32).

[Se5]

J-P. Serre. Eztensions ico.Jaedrique.J, Sem. Th. des Nombres, Bor· deaux, 1 979-80, expose 19 (=C.P. no. 123).

[Se6]

5,

J·P. Serre. L 'in'Variant de Witt de la fO Ym e Tr (:z: 2 ), Comm. Ma.th. Helv. 59 (1984), 651-679. (=C.P. no. 131).

[ Se7]

J-P. Serre. Modular forms of weight one and Galois repre.lenta· nOM, Algebraic Number Fields ( A. Frohlich edit.) Acad. Press, 1977, 193·268 . (=C.P. no. 110).

[Se8]

J-P. Serre. Groupe!J de Galois .Jur Q, Sem. Bourbaki 1 987- 1988] no. 689.

[Se9]

J-P. Serre. Lectures on the Mordell-Weil theorem, translated and edited by M. Brown from notes by M. Waldschmidt, Vieweg­ Verlag, 1989.

[SelO]

J-P. Serre. Specialisation de!J elements de Br2(Q(Th . . . , Tn) C.R. Aca.d. Sci. Paris 311 (1990), 397�402 .

,

[Sel l]

J-P. Serre. Rcvctement.J de courbc.J algebrique.J, Sem. Bourbakit 1991- 1992, no. 749.

[Sell

E.S. S elmer. On the irreducibility of certain trinomiau, Ma.th. Scand. 4 ( 1 95 6) , 287·302.

Bibliography

[SGAl)

[Shall

[Sha2]

115 A. Grothendieck. Revetement etales et groupe fondamental, Lect. Notes in Math. 224, Spnnger- Verlag� 1971. L R. Shafarevich. Construction of fields of algebraic number.. with given JDlvable Gdois group, Izv. Aka4. Nauk S S SR 18 ( 1 954 ) , 525-578 ( = Amer. Math. Transl. 4 (1956), 185-237 = C. P., 139191). I.R. Shafarevich.

Th.e embedding problem for Jplit eztensions,

Dokl. Abd. Nauk SSSR 120 (1958). 1217-1219 ( = C . P . , 205-207).

the factor, of 4 deJcending centroJ .. erie.. (in (1989)� 1 1 4- 1 17.

[Sha.3]

I . R. Shafarevich. On

[Shih 1 ]

K-y. Shih. On the construction of Galou e:r:tensions of fu,nction fieltb and number fields, Math. Ann. 207 (1974), 99-120.

[S hi h2 ]

K-y. Shih . p-division point4 Math. 36 ( 1978), 1 13-129.

[Shim]

Russian), Mat.Zam. 45

on

certain elliptic

G. Shimura. Introduction to the arithmetic theory of automor· phic functions, Publ. Math.Soc.Japan Princeton Uni verJity P res s , 1971 .

[Sie]

[Sw l]

CU",e3, Compo

11,

Iwanami Shoten and

C.L. Siegel. Topics in complex function theory. vol.

2,

Interscience, N ew York, 1971.

R. Swan.

Wiley­

Invariant rational fu,nctions and a problem of Steenrod, ( 1969). 148- 158.

Invent. Math. 7 [Sw2]

R. Swan.

Noether '.t problem in Ga.lois theory,

(J.D. 1983.

Bryn Ma.wr Verlag, [Th1]

J.G.

Springer­

non.-d1Ullity theorem for finite group.., 14 (1970), 1-4.

J.G. Thompson. wh.e re

[Th 3]

2 1-40,

Thompson. A

Algebra [Th2]

Emmy Noether in

Sally and B . Srinivasan edit . ),

K�

Some fini.te group.. wh.ich appear CIS J. of Algebra. 89 (1984), 437-499.

J. of

Ga.l(L/ K),

Q (.u.. ),

J.G. Thomp son.

PSLa and Galois group.. over Q, Proc. Rutgers 1983-1984, Cambridge Univ. Press, 1984. 309-

group theory year

319. [Vi]

N. Vila. On centrAl e:ttensions of A,.

Arch. Ma.th .

44 (1985), 424-437.

CIS

Gdois group over

Q,

Bibliography

116

[Vol ]

V.E. Voskresenskii . Bira.tional propertie' of linear algebraic groUp', Izv. Akad. Nauk SS SR 34 ( 1970), 3- 19 (= Math. USSR Izv. 4 ( 1970), 1- 1 7).

[Vo2]

V . E. Voskresenskii. On the que,tion of the ..t",dure of the sub­

fielda of invariants of a cyclic group oj automorphisms of the field

Q (Zt, . . . , Zft), Iz". Akad. Nauk SSSR 34 ( 1 970) ,

366- 375 (=Math.

USSR Izv. 4 ( 1970), 371-380) . [Wa]

S. Wang. A counten:zample to Grunwald ', theorem, Ann. Ma.th. 49 ( 1948 ) , 1008-1 009.

[We]

A. Weil. Adeles and algebrai c groups. Notes by

T.

[Wi] [Z]

Ono, Birkhiuser-Boston, 1982.

M.

Demazure and

H. Wielandt. Finite Permutation Groupst Acad. Press, New York, 1964. O. a.

Zariski.

Pencils

on

an algebrai.c variety and a new proof of

theorem oj Bertini, Trans. A.M.S. 50 (1941 ), 48-70 (=C ,P" 154- 1 76).

I,

Index Abhy-uwll

�aUedure.

permu tatioD �oru., 36

42

abloJut.ely irreducible 'lariet.y. 20

Briq cune,

86

chvact.er pup

(of Il tonn)t e

profinite completion (of' a diac:rete property (WA), 28

property (WWA), 28

Chowt• theorem , 55

property GalT, 35

DaveDpoft..BaJbcntun theorem, lOS

double pup trick. 43

Fiacher-Gri .. MODlter )I. 11

Q-ratioDal (coaj ulacy

Q-ratiODal. ziii

claII) .

ratioulity (of' a coDjulacy

fundameDt.al sroup (Illebraic). 51 GAGA principle, 55

rqulu

(extenlion). 35

raidu..

dua),

85

(of' a quadratic (orm), 8S

rntridiOil of' IC&lan (CuDdor). 21

GaJr property, 35

rigid «(amDy of' CODjUlacy claua). 10

pDerally true (property), 100 poeric (coveriq map). ziv

rilidity, 70

HIlI.Janko croup J2 , 18

Schol... Reicbardt. iheorem. 8

GraueR-Remmert ut.maiOll theorem, 56

Harder'l theorem. e1

SN

•

property. 10

_parable (profizU�e lfouP). 15

leXtic raolveDt.. 25

B...... Witt iDvuiut

(of' a quadratic Corm). IS

(of a Driety).

85

ra�ioul over Q (variety), ziii

Ftattini Clubpoup), 16

Hilbert property

l1'0up). 58

Shafarevicb'l theorem, 8 18

Bilbert'. irreducibility theorem, 25

Shih'l iheorem, 47

ailllature (OD a RiemaDn .udace). 60

Hilhertian (field), 18

�lit (toru.), 6

Irr(P) property. 23

Irreducibility iheorem (HUb ert) , 25

Stiefel-WhitDey cluIa

Jordan'. lemma, 45

drietly rilid (family oC eODjUluy daueI). 70

K-uDirational (variety). 30

torUl, 6

Mestre'. theorem, gg

t.ype (el), 18

Janko &!,ou p Jlt 77

K-rat.ional (cODjulacy cl ..) , 85

IUle aiew iDequalit.y. 103 Milt"tt '. t.heorem, IiU'

Monlte lfouP. 78

MotM C1llIctiOD. 3IiI nonnalilcrt 1 1

SpriDler'. iheorem, 88

(of a quadratic f'ann ) , e6

thin (Iublet of' V(K», 1 8 trace Conn (of' aD etale allebra), 88

type

(C, ).

18

Yena.I (eoverinl map). ziv weak approximation property, 29 weak weak approzimatioD property. 28

117