Low-lying zeros of elliptic curve L-functions: Beyond the ratios conjecture

TL;DR

This paper provides a precise analysis of the low-lying zeros of elliptic curve L-functions for quadratic twists, surpassing the accuracy predicted by the Ratios Conjecture for certain test functions.

Contribution

It offers a highly accurate computation of the 1-level density of low-lying zeros for quadratic twists of elliptic curves, extending beyond existing conjectural predictions.

Findings

Achieved sharper error bounds than the Ratios Conjecture for specific test functions.

Derived explicit formulas for the 1-level density of low-lying zeros.

Enhanced understanding of the distribution of zeros in elliptic curve L-functions.

Abstract

We study the low-lying zeros of L-functions attached to quadratic twists of a given elliptic curve E defined over $\mathbb Q$. We are primarily interested in the family of all twists coprime to the conductor of E and compute a very precise expression for the corresponding 1-level density. In particular, for test functions whose Fourier transforms have sufficiently restricted support, we are able to compute the 1-level density up to an error term that is significantly sharper than the square-root error term predicted by the L-functions Ratios Conjecture.