Large deviations for values of $L$-functions attached to cusp forms in the level aspect

TL;DR

This paper investigates the distribution of automorphic $L$-function values for holomorphic cusp forms with prime level, deriving an asymptotic formula and applying it to estimate large $L$-function values.

Contribution

It introduces a new asymptotic formula for the density of $L$-function values in a cusp form family, advancing understanding of their large deviations.

Findings

Derived an asymptotic density formula for $L$-function values

Estimated probabilities of large $L$-function values

Enhanced understanding of value distribution in the level aspect

Abstract

We study the distribution of values of automorphic $L$-functions in a family of holomorphic cusp forms with prime level. We prove an asymptotic formula for a certain density function closely related to this value-distribution. The formula is applied to estimate large values of $L$-functions.