COHEN-LENSTRA FOR QUADRATIC FUNCTION FIELDS

3

LEMMA 2.2 (Kowalski).

[17,

Prop. 4.7] In the situation of Theorem 2.1, for

B one may take 2IGIB1 where B1 is any number such that for every etale Galois

cover¢: Y ---. S with deg ¢ invertible in IF, we have ac(Y, Qe)

~

deg ¢ · B1.

PROOF. A representation w : G ---. GLn(Qe) induces a lisse E-. sheaf on S,

denoted

:Fu;,

for some finite extension E-. of Qe. By [16, 9.2.6.(4)], forB one may

take 2B2 where

B

2

is any number such that for all

w,

By

[17,

Prop. 4.7], for

B2

one may take IGIB1, where B1 is any number such that

for every etale Galois cover ¢ : Y ---. S with deg ¢ invertible in IF, a c(Y, Qe)

~

deg¢ · B1.

0

3. Cohen-Lenstra for function fields

We introduce some notation necessary for stating our form of the Friedman-

Washington conjecture (Theorem

3.1).

First, we require some notation about the group of symplectic similitudes. Let

N be an odd natural number, and fix a natural number

g.

Let V be a free Z/N-

module ofrank 2g equipped with a symplectic pairing(·,·). We use this as a model

for GSp

29

(Z/N):

GSp

29

(Z/N)

~

GSp(V, (·, ·))

={A E GL(V)I:Jm(A) E (Z/N)x : 't/v, wE V, (Av, Aw)

=

m(A)(v, w)}.

The map A

f---

m(A) is a homomorphism GSp

29

(Z/N)---. (Z/N)x. For r

E

(Z/N)x,

let

GSp~i(Z/N)

=

m-

1(r);

each

GSp~i(Z/N)

is a torsor over Sp29 (Z/N).

If

L

is any finite abelian group annihilated by

N,

let

l{x

E

GSp

29

(Z/N)Cr): ker(x- id) ~

L}l

a(g,r,N,L)

=

I ( / )I

Sp29 Z N

In the special case where

N

is prime and

r

=

1, an explicit formula for

a(g, r, N, L)

is given by

[1,

Lemma 2.2]; see

[12]

for a formula for

n:(1,

r,

N, L)

for arbitrary

N

and

L.

For general

N,

Goursat's lemma lets one reduce the calculation to the case

where N is a prime power. We will see below (Theorem

3.1)

that

n:(g,

IIFI, N, L) is

a good approximation for the proportion of genus

g

quadratic function fields over

IF for which the N-torsion in the class group is isomorphic to

L.

Second, we introduce a family of hyperelliptic curves. For a natural number n,

let

Hn

be the space parametrizing all monic separable polynomials of degree

n.

Let

C9

---.1{29+2

be the relative smooth proper curve whose fiber over

f(x)

E 7i29+

2

(1F)

has affine model y

2

=

f(x).

(Note that every hyperelliptic curve admits such a

model.)

With these preparations, we can state and prove a theorem of Cohen-Lenstra

type for quadratic function fields.