Vol. 14, No. 7, 2020

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
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 ${\mathbb{𝔽}}_{q}$ of odd characteristic, we give an explicit upper bound on the proportion of genus $g$ hyperelliptic curves over ${\mathbb{𝔽}}_{q}$ whose zeta function vanishes at $\frac{1}{2}+it$. Our upper bound is independent of $g$ and tends to $0$ as $q$ grows.

Keywords
nonvanishing, L-functions, function fields, Dirichlet characters
Primary: 11M38