Average values of \(L\)-functions in characteristic two
type: Item
instance of: scholarly article
title: Average values of \(L\)-functions in characteristic two
zbMATH Open document ID: 1217.11107
author: Yenmei J. Chen
published in: Journal of Number Theory
publication date: 2008-12-08T00:00:00Z
review text: Let \(k={\mathbb F}_q(t)\) be the rational function field over the finite field of \(q\) elements. For a non-square polynomial \(M\) in the ring \({\mathbf A} = {\mathbb F}_q[t]\), consider the \({\mathbf A}\)-order \({\mathcal O}_M = {\mathbf A} + {\mathbf A}\sqrt M\) in the ring of integers of the field \(k(\sqrt M)\). Let \(h_M\) denote the class number of \({\mathcal O}_M\). \textit{J.\ Hoffstein} and \textit{M.\ Rosen} [J. Reine Angew. Math. 426, 117--150 (1992; Zbl 0754.11036)] proved a formula for the mean value of \(h_M\), assuming that \(q\) is odd. Here \(M\) varies over monic polynomials of fixed degree. This formula is an analogue of the corresponding formula in the number field case, conjectured by Gauss and proven by Lipschitz and Siegel. The present author extends the formula to the case of even \(q\). To this end, he first proves an analogue of Dirichlet's class number formula for the \(L\)-functions associated to \({\mathcal O}_M\) for even \(q\). A basic difficulty now is that there is no bijection between quadratic fields and their discriminants. Therefore the problem of averaging \(h_M\) over all fundamental discriminants must be left open. However, the author has found a suitable parametrization of the fields which enables him to obtain nice averaging formulas.
Mathematics Subject Classification ID: 11R54
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11R65
zbMATH DE Number: 5379240
zbMATH Keywords: \(L\)-functions of orders
zbMATH Keywords: Quadratic extensions
zbMATH Keywords: Class numbers
zbMATH Keywords: Arithmetic theory of algebraic function fields
zbMATH Keywords: Picard groups of orders
MaRDI profile type: Publication
full work available at URL: j.jnt.2007.12.011
OpenAlex ID: W2014931112
cites work: On class number relations in characteristic two
cites work: Q3996190
cites work: Average values of L-series in function fields.
cites work: Q4075566
cites work: On class number relations over function fields
DOI: 10.1016/J.JNT.2007.12.011
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description: scientific article; zbMATH DE number 5379240