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

About this Title

A. Knightly and C. Li

Publication: Memoirs of the American Mathematical Society
Publication Year: 2013; Volume 224, Number 1055
ISBNs: 978-0-8218-8744-8 (print); 978-1-4704-1006-3 (online)
Published electronically: December 6, 2012
Keywords: Sum formula, Maass forms, Kloosterman sums, Hecke eigenvalues, equidistribution, trace formula
MSC: Primary 11F72, 11F70, 11F41, 11F37, 11F30, 11L05, 11F25, 22E55

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Table of Contents


  • 1. Introduction
  • 2. Preliminaries
  • 3. Bi-$K_\infty$-invariant functions on $\operatorname {GL}_2(\mathbf {R})$
  • 4. Maass cusp forms
  • 5. Eisenstein series
  • 6. The kernel of $R(f)$
  • 7. A Fourier trace formula for $\operatorname {GL}(2)$
  • 8. Validity of the KTF for a broader class of $h$
  • 9. Kloosterman sums
  • 10. Equidistribution of Hecke eigenvalues


We 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. 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.

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