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.
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978-0-8218-8744-8 (9780821887448)
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Schweitzer Klassifikation
A. Knightly, University of Maine, Orono, ME, USA
C. Li, The Chinese University of Hong Kong, China
Introduction Preliminaries
Bi-$K_\infty$-invariant functions on $\operatorname{GL}_2(\mathbf{R})$
Maass cusp forms
Eisenstein series
The kernel of $R(f)$
A Fourier trace formula for $\operatorname{GL}(2)$
Validity of the KTF for a broader class of $h$ Kloosterman sums
Equidistribution of Hecke eigenvalues
Bibliography
Notation index
Subject index
