Kuznetsov's Trace Formula and the Hecke Eigenvalues of Maass Forms

Provides an adelic treatment of the Kuznetsov trace formula as a relative trace formula on $/operatorname{GL}(2)$ over $/mathbf{Q}$. 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.

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The authors give an adelic treatment of the Kuznetsov trace formula as a relative trace formula on $/operatorname{GL}(2)$ over $/mathbf{Q}$. 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.