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£).
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