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
DOI: 10.2140/ant.2020.14.1895
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.

Keywords
nonvanishing, L-functions, function fields, Dirichlet characters
Mathematical Subject Classification 2010
Primary: 11M38
Milestones
Received: 22 February 2019
Revised: 18 December 2019
Accepted: 6 February 2020
Published: 18 August 2020
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