Memoirs of the American Mathematical Society 2013; 132 pp; softcover Volume: 224 ISBN10: 0821887440 ISBN13: 9780821887448 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 SatoTate 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
