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

The authors give an adelic treatment of the Kuznetsov trace formula as a relative trace formula on $operatornameGL(2)$ over $mathbfQ$. The result is a variant which incorporates a Hecke eigenvalue in addition to two Fourier coefficients on the spectral side. The authors include a proof of a Weil bound for the generalized twisted Kloosterman sums which arise on the geometric side. As an application, they 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.