Electronic ISBN:  9781470410063 
Product Code:  MEMO/224/1055.E 
List Price:  $73.00 
MAA Member Price:  $65.70 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 224; 2013; 132 ppMSC: Primary 11; 22;
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 SatoTate measure in the limit as the level goes to infinity.

Table of Contents

Chapters

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


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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 SatoTate measure in the limit as the level goes to infinity.

Chapters

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