Vanishing of Quartic and Sextic Twists of $L$-functions

TL;DR

This paper conjectures asymptotic estimates for the vanishing of certain elliptic curve $L$-functions twisted by quartic and sextic characters, supported by numerical evidence and analysis of Gauss sum equidistribution.

Contribution

It introduces conjectures for the distribution of vanishing $L$-values for quartic and sextic twists, extending previous work to composite orders and analyzing Gauss sum equidistribution.

Findings

Conjectured asymptotics for vanishing of $L$-functions with quartic and sextic twists.

Numerical evidence supporting the conjectures.

Analysis of the equidistribution rate of Gauss sums for these characters.

Abstract

Let $E$ be an elliptic curve over $\mathbf{Q}$. We conjecture asymptotic estimates for the number of vanishings of $L(E,1,\chi)$ as $\chi$ varies over all primitive Dirichlet characters of orders 4 and 6, subject to a mild hypothesis on $E$. Our conjectures about these families come from conjectures about random unitary matrices as predicted by the philosophy of Katz-Sarnak. We support our conjectures with numerical evidence. Earlier work by David, Fearnley and Kisilevsky formulates analogous conjectures for characters of any odd prime order. In the composite order case, however, we need to justify our use of random matrix theory heuristics by analyzing the equidistribution of the squares of normalized Gauss sums. Along the way we introduce the notion of totally order $\ell$ characters to quantify how quickly quartic and sextic Gauss sums become equidistributed. Surprisingly, the rate of equidistribution in the full family of quartic (sextic, resp.) characters is much slower than in the sub-family of totally quartic (sextic, resp.) characters. A conceptual explanation for this phenomenon is that the full family of order $\ell$ twisted elliptic curve $L$-functions, with $\ell$ even and composite, is a mixed family with both unitary and orthogonal aspects.