Nonvanishing of hyperelliptic zeta functions 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.

References in corpus (2)

• Vanishing of Hyperelliptic L-functions at the Central Point
• Cohen Lenstra Heuristics for Étale Group Schemes and Symplectic Pairings

Cited by in corpus (2)

• Moments of quadratic Dirichlet $L$-functions over Function Fields
• Exceptional biases in counting primes over functions fields