Conjectures for the integral moments and ratios of L-functions over function fields

Abstract

We extend to the function field setting the heuristic previously developed, by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments and ratios of L-functions defined over number fields. Specifically, we give a heuristic for the moments and ratios of a family of L-functions associated with hyperelliptic curves of genus g over a fixed finite field Fq in the limit as g→∞. Like in the number field case, there is a striking resemblance to the corresponding formulae for the characteristic polynomials of random matrices. As an application, we calculate the one-level density for the zeros of these L-functions.