4 JEFFREY D. ACHTER THEOREM 3.1. Let g be a natural number, let N be an odd natural number, and let lF be a sufficiently large finite field in which IGSp29 (Z/N)I is invertible. Then l l{f(x) E 1i2g+2(1F): Jac(C9,t)(JF)[N] ~ L}l _ ( IJFI N L)l l1i2g+2(1F)I a g, ' ' 2(2g + 1)!1GSp29 (Z/N)I ~ . PROOF. Consider the etale sheaf :FN = Jac(C9)[N] --+ 1i2g+2, which corre- sponds (after fixing a basepoint fj) to a representation p : n 1 (1i2g+2, fj) --+ Aut(:F N,iJ) ~ GL29 (Z/N). Computing the proportion of points f E 1i29+2(1F) for which Jac(C9,t)[N] ~Lis the same as computing the proportion of points f E 1i29+2(JF) for which ker(p(Fr f,JF) - id) ~ L the latter task is accomplished using Katz's the- orem 2.1. In the notation of (2.1), ageom ~ Sp29 (Z/N). If N is prime, this is attributed to an unpublished work of J.K. Yu (see [8, 2.4]) and is proved in [2, Thm. 3.4] and in [13, Thm. 4.1]. The case where N is a prime power follows formally from this [2, Cor. 3.5], while the case of general N (still prime to the characteristic) follows from Goursat's lemma [17, Cor. 2.6]. By Lemma 2.2, forB in (2.2) one may take IGSp29(Z/N)IB2, where B2 is any number such that for every etale Galois cover ¢ : Y --+ 1i29+2 with deg ¢ invertible in lF, ac(Y, Q£) :-:=: deg ¢ · B2. We provide an explicit value for B2 as follows. Let z1, · · · , z29+2 be coordinates on affine space A29+2, and let H29+2 = A 29+2 - Uiij ( Zi = Zj) be the complement of the fat diagonal. It is an irreducible, etale 829+2-cover of 1i29+2 the geometric fiber over f E 1i2g+2 (JF) is the set of labelings of the roots of f. Let ¢ : Y --+ 1i29+2 be an etale Galois cover with automorphism group G. Trivially one has ac(Y, Q£) :-:=: acQ'" XH. 29 +2 H2g+2, Q£). Let r be the number of con- nected components of Y X11. 29 +2 1i2g+2· Each component is isomorphic to a K-cover Z of H2g+2, where K is a subgroup of G of index r. By Lemma 3.3, ac(Z, Q£) :-:=: IKI · ac(H2g+2, Q£). Since H2g+2 is smooth, ac(H2g+2, Q£) = a(H2g+2, Q£). The calculation a(H29+2,Q£) = (2g + 1)!, originally due to Arnol'd [4], may also be recovered from equation (3.2) below. D The space 1i29+2 introduced in the proof of Theorem 3.1 is the braid arrange- ment studied by Arnol' d. In Lemma 3.3 we prove a case of [17, Prop. 4.5] optimized for hyperplane arrangements. Let T--+ S be a Galois cover of smooth varieties, and let SH C S be a suitably generic hyperplane section. The strategy of [17, Prop. 4.5], which is a refinement of the argument of [15], is to relate the Betti numbers ofT to those of TH = T x s S H. This inductive method produces explicit upper bounds for the Betti numbers ofT, but they tend to be pessimistically large. For example, the bounds obtained from [17] for the Betti numbers of 1in are much larger than n n. In the special case where S is a hyperplane arrangement, we can replace the (abstract) Lefschetz theorem with an explicit calculation of a(S, Q£)- a(SH, Q£). Moreover, SH is itself a hyperplane arrangement, so that we may use induction on the dimension of S without leaving the class of hyperplane arrangements. Let V / k be an n-dimensional vector space over an algebraically closed field, and let A be a finite collection of hyperplanes in V. The complement of this
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