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

Abstract

We study the distribution of extreme values of automorphic $L$-functions in a family of holomorphic cusp forms with prime levels. First, we prove an asymptotic formula for a certain density function closely related to this value-distribution. Then we apply it to estimate the probabilities of large deviations for values of automorphic $L$-functions.