Low-lying zeros in families of elliptic curve $L$-functions over function fields

TL;DR

This paper studies the distribution of low-lying zeros in families of elliptic curve $L$-functions over function fields, revealing orthogonal symmetry and refining previous results for quadratic twists, while pioneering analysis for cubic twists.

Contribution

It provides explicit formulas for trace expectations, confirms orthogonal symmetry, and extends analysis to cubic twists, which was not previously explored.

Findings

Elliptic curve families exhibit orthogonal symmetry.

Refined lower order terms in quadratic twist families.

First analysis of low-lying zeros in cubic twist families.

Abstract

We investigate the low-lying zeros in families of $L$-functions attached to quadratic and cubic twists of elliptic curves defined over $\mathbb{F}_q(T)$. In particular, we present precise expressions for the expected values of traces of high powers of the Frobenius class in these families with a focus on the lower order behavior. As an application we obtain results on one-level densities and we verify that these elliptic curve families have orthogonal symmetry type. In the quadratic twist families our results refine previous work of Comeau-Lapointe. Moreover, in this case we find a lower order term in the one-level density reminiscent of the deviation term found by Rudnick in the hyperelliptic ensemble. On the other hand, our investigation is the first to treat these questions in families of cubic twists of elliptic curves and in this case it turns out to be more complicated to isolate lower order terms due to a larger degree of cancellation among lower order contributions.