Convergence to the Plancherel measure of Hecke Eigenvalues

TL;DR

This paper provides improved estimates for how quickly Hecke eigenvalues of certain modular forms converge to the Plancherel measure, with applications to random walks on Ramanujan graphs and bounds on modular form multiplicities.

Contribution

It offers sharper uniform convergence estimates for Hecke eigenvalues and applies these results to problems in graph theory and modular form multiplicities.

Findings

Enhanced convergence rate bounds for Hecke eigenvalues

Determined sharp cutoff for non-backtracking random walks on Ramanujan graphs

Bounded multiplicities of modular forms with coefficients in number fields

Abstract

We give improved uniform estimates for the rate of convergence to Plancherel measure of Hecke eigenvalues of holomorphic forms of weight 2 and level N. These are applied to determine the sharp cutoff for the non-backtracking random walk on arithmetic Ramanujan graphs and to Serre's problem of bounding the multiplicities of modular forms whose coefficients lie in number fields of degree d.