Average values of L-functions in even characteristic

Abstract

Let k=Fq(T) be the rational function field over a finite field Fq, where q is a power of 2. In this paper we solve the problem of averaging the quadratic L-functions L(s,χu) over fundamental discriminants. Any separable quadratic extension K of k is of the form K=k(xu), where xu is a zero of X2+X+u=0 for some u∈k. We characterize the family I (resp. F, F′) of rational functions u∈k such that any separable quadratic extension K of k in which the infinite prime ∞=(1/T) of k ramifies (resp. splits, is inert) can be written as K=k(xu) with a unique u∈I (resp. u∈F, u∈F′). For almost all s∈C with Re(s)≥12, we obtain the asymptotic formulas for the summation of L(s,χu) over all k(xu) with u∈I, all k(xu) with u∈F or all k(xu) with u∈F′ of given genus. As applications, we obtain the asymptotic mean value formulas of L-functions at s=12 and s=1 and the asymptotic mean value formulas of the class number hu or the class number times regulator huRu.