memo_has_moved_text();Kuznetsov’s trace formula and the Hecke eigenvalues of Maass forms

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)
DOI: http://dx.doi.org/10.1090/S0065-9266-2012-00673-3
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

View full volume PDF

View other years and numbers:

Chapters

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

Abstract

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.