COHEN-LENSTRA FOR QUADRATIC FUNCTION FIELDS 3 LEMMA 2.2 (Kowalski). [17, Prop. 4.7] In the situation of Theorem 2.1, for B one may take 2IGIB1 where B1 is any number such that for every etale Galois cover¢: Y ---. S with deg ¢ invertible in IF, we have ac(Y, Qe) ~ deg ¢ · B1. PROOF. A representation w : G ---. GLn(Qe) induces a lisse E-. sheaf on S, denoted :Fu , for some finite extension E-. of Qe. By [16, 9.2.6.(4)], forB one may take 2B2 where B 2 is any number such that for all w, By [17, Prop. 4.7], for B2 one may take IGIB1, where B1 is any number such that for every etale Galois cover ¢ : Y ---. S with deg ¢ invertible in IF, a c(Y, Qe) ~ deg¢ · B1. 0 3. Cohen-Lenstra for function fields We introduce some notation necessary for stating our form of the Friedman- Washington conjecture (Theorem 3.1). First, we require some notation about the group of symplectic similitudes. Let N be an odd natural number, and fix a natural number g. Let V be a free Z/N- module ofrank 2g equipped with a symplectic pairing(·,·). We use this as a model for GSp29 (Z/N): GSp29 (Z/N) ~ GSp(V, (·, ·)) ={A E GL(V)I:Jm(A) E (Z/N)x : 't/v, wE V, (Av, Aw) = m(A)(v, w)}. The map A f--- m(A) is a homomorphism GSp29 (Z/N)---. (Z/N)x. For r E (Z/N)x, let GSp~i(Z/N) = m-1(r) each GSp~i(Z/N) is a torsor over Sp29 (Z/N). If L is any finite abelian group annihilated by N, let l{x E GSp29 (Z/N)Cr): ker(x- id) ~ L}l a(g,r,N,L) = I ( / )I Sp29 Z N In the special case where N is prime and r = 1, an explicit formula for a(g, r, N, L) is given by [1, Lemma 2.2] see [12] for a formula for n:(1, r, N, L) for arbitrary N and L. For general N, Goursat's lemma lets one reduce the calculation to the case where N is a prime power. We will see below (Theorem 3.1) that n:(g, IIFI, N, L) is a good approximation for the proportion of genus g quadratic function fields over IF for which the N-torsion in the class group is isomorphic to L. Second, we introduce a family of hyperelliptic curves. For a natural number n, let Hn be the space parametrizing all monic separable polynomials of degree n. Let C9 ---.1{29+2 be the relative smooth proper curve whose fiber over f(x) E 7i29+ 2 (1F) has affine model y2 = f(x). (Note that every hyperelliptic curve admits such a model.) With these preparations, we can state and prove a theorem of Cohen-Lenstra type for quadratic function fields.
Previous Page Next Page