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 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

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


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
= 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 (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.


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