Average values of L-functions in even characteristic

Let k = F-q(T) be the rational function field over a finite field F-q, where q is a power of 2. In this paper we solve the problem of averaging the quadratic L-functions L(s, chi(u)) over fundamental discriminants. Any separable quadratic extension K of k is of the form K = k(x(u)), where x(u) is a zero of X-2 + X + u = 0 for some u is an element of k. We characterize the family I (resp. F, F') of rational functions u is an element of k such that any separable quadratic extension K of k in which the infinite prime infinity = (1/T) of k ramifies (resp. splits, is inert) can be written as K = k(x(u)) with a unique u is an element of I (resp. u is an element of F, u is an element of F'). For almost all s is an element of C with Re(s) >= 1/2, we obtain the asymptotic formulas for the summation of L(s,chi(u)) over all k(x(u)) with u is an element of I, all k(x(u)) with u is an element of F or all k(x(u)) with u is an element of F' of given genus. As applications, we obtain the asymptotic mean value formulas of L-functions at s = 1/2 and s = 1 and the asymptotic mean value formulas of the class number h(u) or the class number times regulator h(u)R(u). (C) 2017 Elsevier Inc. All rights reserved.
Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
Issue Date
2018-05
Language
English
Article Type
Article
Keywords

MEAN-VALUE; HYPERELLIPTIC ENSEMBLE; FUNCTION-FIELDS; L-SERIES; CHI); MOMENTS; L(1/2

Citation

JOURNAL OF NUMBER THEORY, v.186, pp.269 - 303

ISSN
0022-314X
DOI
10.1016/j.jnt.2017.10.006
URI
http://hdl.handle.net/10203/240584
Appears in Collection
MA-Journal Papers(저널논문)
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