Nonvanishing of hyperelliptic zeta functions over finite fields

Ellenberg, Jordan S.; Li, Wanlin; Shusterman, Mark

doi:10.2140/ant.2020.14.1895

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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 $t \in ℝ$ and a finite field $𝔽_{q}$ of odd characteristic, we give an explicit upper bound on the proportion of genus $g$ hyperelliptic curves over $𝔽_{q}$ whose zeta function vanishes at $\frac{1}{2} + i t$ . 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

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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

Vol. 14, No. 7, 2020