Vanishing of Hyperelliptic L-functions at the Central Point

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.

References in corpus (2)

• Character formula for Weil representations in terms of Frobenius traces
• Character formula for conjugacy classes in a coset

Cited by in corpus (7)

• Chebyshev's bias for products of irreducible polynomials
• Roots of $L$-functions of characters over function fields, generic linear independence and biases
• Nonvanishing of hyperelliptic zeta functions over finite fields
• Limiting properties of the distribution of primes in an arbitrarily large number of residue classes
• Moments of quadratic Dirichlet $L$-functions over Function Fields
• One-level density and non-vanishing for cubic $L$-functions over the Eisenstein field
• Exceptional biases in counting primes over functions fields