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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