The mean values of cubic L-functions over function fields

David, Chantal; Florea, Alexandra; Lalín, Matilde

doi:10.2140/ant.2022.16.1259

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The mean values of cubic $L$-functions over function fields

Chantal David, Alexandra Florea and Matilde Lalín

Vol. 16 (2022), No. 5, 1259–1326

DOI: 10.2140/ant.2022.16.1259

Abstract

We obtain an asymptotic formula for the mean value of $L$ -functions associated to cubic characters over $𝔽_{q} [T]$ . We solve this problem in the non-Kummer setting when $q \equiv 2 (\mod 3)$ and in the Kummer setting when $q \equiv 1 (\mod 3)$ . In the Kummer setting, the mean value over the complete family of cubic characters was never addressed in the literature (over number fields or function fields). The proofs rely on obtaining precise asymptotics for averages of cubic Gauss sums over function fields, which can be studied using the pioneer work of Kubota. In the non-Kummer setting, we display some explicit (and unexpected) cancellation between the main term and the dual term coming from the approximate functional equation of the $L$ -functions. Exhibiting the cancellation involves evaluating sums of residues of a variant of the generating series of cubic Gauss sums.

Keywords

moments over function fields, cubic twists, nonvanishing

Mathematical Subject Classification

Primary: 11M06, 11M38, 11R16

Secondary: 11R58

Milestones

Received: 19 May 2021

Accepted: 26 July 2021

Published: 16 August 2022

Authors

Chantal David
Department of Mathematics and Statistics Concordia University Montreal, QC Canada

Alexandra Florea
Department of Mathematics University of California Irvine, CA United States

Matilde Lalín
Département de mathématiques et de statistique Université de Montréal Montreal, QC Canada

Vol. 16, No. 5, 2022