Conditional lower bounds on the distribution of central values in families of $L$-functions

TL;DR

This paper presents a general method to strengthen lower bounds on the non-vanishing of central $L$-values, demonstrating that most such values align with conjectured typical sizes, with applications to quadratic twists of elliptic curves.

Contribution

It introduces a principle to refine lower bounds from one-level density studies, showing most $L$-values match conjectured size distributions, exemplified in elliptic curve twists.

Findings

Most $L$-values have the typical size conjectured by Keating and Snaith.

The technique applies to quadratic twists of elliptic curves.

Similar results extend to other families studied by Iwaniec, Luo, and Sarnak.

Abstract

We establish a general principle that any lower bound on the non-vanishing of central $L$-values obtained through studying the one-level density of low-lying zeros can be refined to show that most such $L$-values have the typical size conjectured by Keating and Snaith. We illustrate this technique in the case of quadratic twists of a given elliptic curve, and similar results would hold for the many examples studied by Iwaniec, Luo, and Sarnak in their pioneering work on $1$-level densities.