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.
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