Vanishing of Hyperelliptic L-functions at the Central Point

TL;DR

This paper establishes a lower bound on the number of quadratic Dirichlet L-functions over function fields that vanish at the central point, contrasting with the conjecture over rational numbers, using geometric interpretations.

Contribution

It introduces a geometric approach to bound the number of vanishing L-functions over function fields, providing new insights into their distribution.

Findings

Lower bound on vanishing L-functions over function fields

Geometric interpretation of vanishing as maps to abelian varieties

Contrasts with conjectured non-vanishing over rationals

Abstract

We obtain a lower bound on the number of quadratic Dirichlet L-functions over the rational function field which vanish at the central point $s = 1/2$. This is in contrast with the situation over the rational numbers, where a conjecture of Chowla predicts there should be no such L-functions. The approach is based on the observation that vanishing at the central point can be interpreted geometrically, as the existence of a map to a fixed abelian variety from the hyperelliptic curve associated to the character.