Kuznetsov's trace formula and the Hecke eigenvalues of Maass forms

TL;DR

This paper develops an adelic version of Kuznetsov's trace formula for GL(2) over Q, incorporating Hecke eigenvalues, and proves their equidistribution at fixed primes as the level increases.

Contribution

It introduces a new adelic formulation of the Kuznetsov trace formula that includes Hecke eigenvalues and proves their equidistribution relative to Sato-Tate measure.

Findings

Hecke eigenvalues become equidistributed as level increases.

A proof of Weil bound for twisted Kloosterman sums.

New adelic trace formula incorporating Hecke eigenvalues.

Abstract

We give an adelic treatment of the Kuznetsov trace formula as a relative trace formula on GL(2) over Q. The result is a variant which incorporates a Hecke eigenvalue in addition to two Fourier coefficients on the spectral side. We include a proof of a Weil bound for the generalized twisted Kloosterman sums which arise on the geometric side. As an application, we show that the Hecke eigenvalues of Maass forms at a fixed prime, when weighted as in the Kuznetsov formula, become equidistributed relative to the Sato-Tate measure in the limit as the level goes to infinity.