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Kuznetsov's Trace Formula and the Hecke Eigenvalues of Maass Forms
A. Knightly, University of Maine, Orono, ME, and C. Li, The Chinese University of Hong Kong, China
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Memoirs of the American Mathematical Society
2013; 132 pp; softcover
Volume: 224
ISBN-10: 0-8218-8744-0
ISBN-13: 978-0-8218-8744-8
List Price: US$73 Member Price: US$58.40
Order Code: MEMO/224/1055

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.

• 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