Vol. 14, No. 7, 2020

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Nonvanishing of hyperelliptic zeta functions over finite fields

Jordan S. Ellenberg, Wanlin Li and Mark Shusterman

Vol. 14 (2020), No. 7, 1895–1909
Abstract

Fixing t and a finite field 𝔽q of odd characteristic, we give an explicit upper bound on the proportion of genus g hyperelliptic curves over 𝔽q whose zeta function vanishes at 1 2 + it. Our upper bound is independent of g and tends to 0 as q grows.

An errata was submitted on 27 Aug 2021 and posted online on 1 Sep 2021.

Keywords
nonvanishing, L-functions, function fields, Dirichlet characters
Mathematical Subject Classification 2010
Primary: 11M38
Supplementary material

Errata

Milestones
Received: 22 February 2019
Revised: 18 December 2019
Accepted: 6 February 2020
Published: 18 August 2020

Errata: 1 September 2021

Authors
Jordan S. Ellenberg
Department of Mathematics
University of Wisconsin
Madison, WI
United States
Wanlin Li
Department of Mathematics
MIT
Cambridge, MA
United States
Mark Shusterman
Department of Mathematics
University of Wisconsin
Madison, WI
United States