#### 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 $t\in ℝ$ 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