**Memoirs of the American Mathematical Society**

2013;
132 pp;
Softcover

MSC: Primary 11; 22;

Print ISBN: 978-0-8218-8744-8

Product Code: MEMO/224/1055

List Price: $73.00

AMS Member Price: $58.40

MAA member Price: $65.70

**Electronic ISBN: 978-1-4704-1006-3
Product Code: MEMO/224/1055.E**

List Price: $73.00

AMS Member Price: $58.40

MAA member Price: $65.70

# Kuznetsov’s Trace Formula and the Hecke Eigenvalues of Maass Forms

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*A. Knightly; C. Li*

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.

#### Table of Contents

# Table of Contents

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

- Chapter 1. Introduction 18 free
- Chapter 2. Preliminaries 714
- Chapter 3. Bi-𝐾_{∞}-invariant functions on 𝐺𝐿₂(𝐑) 1320
- Chapter 4. Maass cusp forms 2330
- Chapter 5. Eisenstein series 3340
- Chapter 6. The kernel of 𝑅(𝑓) 5158
- Chapter 7. A Fourier trace formula for 𝐺𝐿(2) 6572
- Chapter 8. Validity of the KTF for a broader class of ℎ 8390
- Chapter 9. Kloosterman sums 109116
- Chapter 10. Equidistribution of Hecke eigenvalues 121128
- Bibliography 125132