Conditional lower bounds on the distribution of central values: the case of modular forms

TL;DR

This paper investigates the distribution of central values of L-functions associated with modular forms, showing they approximately follow a normal distribution with parameters predicted by the Keating-Snaith conjecture.

Contribution

It extends the understanding of L-function value distributions to the level aspect of modular forms, providing new probabilistic results and lower bounds.

Findings

Logarithms of central L-values follow a normal distribution.

Mean and variance match Keating-Snaith conjecture predictions.

Results apply to families of modular forms with increasing level.

Abstract

Radziwill and Soundararajan unveiled a connection between low-lying zeros and central values of $L$-functions, which they instantiated in the case of quadratic twists of an elliptic curve. This paper addresses the case of the family of modular forms in the level aspect, and proves that the logarithms of central values of associated L-functions approximately distribute along a normal law with mean -(1/2)log log c(f) and variance log log c(f), where c(f) is the analytic conductor of f, as predicted by the Keating-Snaith conjecture.