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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

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
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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 Sato-Tate measure in the limit as the level goes to infinity.

Table of Contents

  • 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
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