6 JEFFREY D. ACHTER The proof of Lemma 3.3 is modeled closely on [17, Prop. 4.5], and we focus on the differences. LEMMA 3.3. Let A be a hyperplane arrangement in an n-dimensional vector space V over an algebraically closed field k, and let¢: Y -t M(A) be an irreducible etale Galois cover of degree m which is invertible ink. Then o"c(Y) ::: mo"c(A). PROOF. By Poincare duality, it suffices to prove the analogous result for the sum of Betti numbers a(Y, Q£). Since m is invertible in k, the cover ¢ is tamely ramified on the boundary of M(A) in V. A result of Deligne and Lusztig [14, 2.6, Cor. 2.8] shows that x(Y, Qt) = mx(M(A), Q£). Our proof is by induction on dim V. If n = 1, let b1 = h1(M(A), Q£). Then x(M(A), Q£) = 1 - b1, so that x(Y, Qt) = m · (1 - b1), and a(Y, Q£) = 1 + (1 + m(b1- 1)) ::: m · (1 + b1) = ma(M(A), Q£). Now assume that the lemma holds for any arrangement in a vector space of dimension n - 1. Using [17, Prop. 4.6], one may choose a hyperplane H C V, generic with respect to A, such that the pullback YH -t M(AH) is an irreducible Galois cover of M(AH ). As in the proof of [17, Prop. 4.5], we find that a(Y, Qt) ::: m (( -1tx(M(A), Q£) + ( -1)n-1x(M(AH), Qt)) + a(YH, Q£). By Lemma 3.2, the first term on the right-hand side is simply m · hn(M(A), Q£), while by the inductive hypothesis the other term is at most m · a(M(AH),Q£). A second application of Lemma 3.2 shows that a(M(AH),Qt) = a(M(A),Qt)- hn(M(A), Q£). Taken together, this shows that a(Y, Q£) ::: m · a(M(A), Q£). 0 References [1] Jeffrey D. Achter, The distribution of class groups of function fields, J. Pure and Appl. Algebra 204 (2006), no. 2, 316-333. [2] Jeffrey D. Achter and Rachel J. Pries, The integral monodromy of hyperelliptic and trielliptic curves, Math. Ann. 338 (2007), no. 1, 187-206. [3] Leonard M. Adleman and Ming-Deh A. Huang, Primality testing and abelian varieties over finite fields, Springer-Verlag, Berlin, 1992. [4] V. I. Arnol'd, The cohomology ring of the group of dyed braids, Mat. Zametki 5 (1969), 227-231. [5] M. Bauer, M. J. Jacobson, Jr., Y. Lee, and R. Scheidler, Construction of hyperelliptic function fields of high three-rank, Math. Comp. 77 (2008), no. 261, 503-530 (electronic). [6] Anders Bjorner and Torsten Ekedahl, Subspace arrangements over finite fields: cohomological and enumerative aspects, Adv. Math. 129 (1997), no. 2, 159-187. [7] David A. Cardon and M. Ram Murty, Exponents of class groups of quadratic function fields over finite fields, Canad. Math. Bull. 44 (2001), no. 4, 398-407. [8] Nick Chavdarov, The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy, Duke Math. J. 87 (1997), no. 1, 151-180. [9] Henri Cohen and Hendrik W. Lenstra, Jr., Heuristics on class groups of number fields, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 33--62. [10] Eduardo Friedman and Lawrence C. Washington, On the distribution of divisor class groups of curves over a finite field, Theorie des nombres (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 227-239. [ll] Christian Friesen, Class group frequencies of real quadratic function fields: the degree 4 case, Math. Comp. 69 (2000), no. 231, 1213-1228. [12] Ernst-Ulrich Gekeler, The distribution of group structures on elliptic curves over finite prime fields, Doc. Math. 11 (2006), ll9-142 (electronic).
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