# Nonvanishing of hyperelliptic zeta functions over finite fields

## TL;DR

This paper provides an explicit upper bound on the proportion of hyperelliptic curves over finite fields whose zeta functions vanish at a specific point, showing the proportion diminishes as the field size increases.

## Contribution

It establishes a bound on the vanishing of hyperelliptic zeta functions at a fixed point, independent of genus, with the bound decreasing as the finite field size grows.

## Key findings

- Upper bound on vanishing proportion is independent of genus
- Proportion tends to zero as field size increases
- Results apply to hyperelliptic curves over finite fields

## Abstract

Fixing $t \in \mathbb{R}$ 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.