2

JEFFREY D. ACHTER

computational [5, 11] and analytic [7, 22] methods have been brought to bear on

the distribution of class groups of hyperelliptic function fields. Roughly speaking,

these works produces families of such fields whose class numbers are divisible by a

given prime £. While these families are infinite, they account for a proportion of

polynomials which

vanishes

as IJFI increases.

In this note, we explain how deep equidistribution results due to Katz [16,

Chapter 9] yield the solution of the Friedman-Washington conjecture. Moreover,

we take advantage of recent refinements to Katz's method by Kowalski

[17]

to

give explicit bounds on the error terms which arise, by bounding the £-adic Betti

numbers of etale covers of hyperplane arrangements.

It's a pleasure to thank Kristin Lauter and Ken Ribet for organizing this work-

shop, and Rachel Pries for comments on this note.

2. Equidistribution

Let S/lF be a geometrically irreducible variety, and let

i}

---t

S be a geometric

point. Let Ggeom be a finite group. An irreducible etale Ggeom_cover of St cor-

responds to a surjective homomorphism pgeom : rr1 (St,r;)

---4

Ggeom. To a point

s

E

S

(JF) corresponds a Frobenius element Fr

8

/IF in

1r1 (

S),

well-defined up to conju-

gacy. Katz proves a strong Chebotarev-type theorem, which states that the images

of these Frobenius elements under

p

are equidistributed.

THEOREM

2.1 (Katz). [16, Thm. 9.7.13]

Let

S/lF

be a smooth geometrically

irreducible variety, and let

i}

---t

S be a geometric point. Suppose we are given a

commutative diagram

(2.1)

1 ----- rrfeom (

S,

i]) -----

11"1 (

S,

i]) -----

Gal(lF)

~

Z -----

1

!

lp

!

1 ---

Ggeom ----

G ___

m __

~

r

----1

where

r

is abelian, G is finite and

IGI

is invertible in

lF.

Let

'Y

E

r

be the image of

the inverse of the Probenius substitution x

f---7

xi1FI. There exists a constant B such

that if C

C

G is stable under conjugation and if

JF' /lF

is an extension of degree n,

then

(2.2)

lj{s

E

S(JF'): p(Frs/IF')

E

C}j_ jCnGbnlJI

__!!__

IS(JF')I IGgeoml

JiiF'j'

where

Q('Yn)

=

m- 1

('Yn).

One can calculate an effective value for B in (2.2) in terms of certain £-adic

Betti numbers. If

X

is a variety over a field

k

in which a rational prime £ is

invertible, the

ith

£-adic Betti number of

X

is

hi(X,Qt)

:=

dimHi(Xii:,Q£). The

sum of these numbers is a(

X,

Q£), and the £-adic Euler characteristic of

X

is the

alternating sum

x(X,Qe)

:=

I::i(-1)ihi(X,Qt)·

We will also need to use the Betti numbers with compact support

h~(X,

Q£)

:=

dimH~(Xk,

Q£). The sum of these compact Betti numbers of

X

is

ac(X,

Q£), and

the alternating sum of these numbers is

Xc(X,

Qe). If

X

is smooth, then Poincare

duality yields the equality

a( X,

Q£)

= ac(X,

Q£).