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

Intro -- Contents -- Chapter 1. Introduction -- 1.1. Some history -- 1.2. Overview of the contents -- 1.3. Acknowledgements -- Chapter 2. Preliminaries -- 2.1. Notation and Haar measure -- 2.2. Characters and Dirichlet -functions -- Chapter 3. Bi- _{∞}-invariant functions on ₂( ) -- 3.1. Several guises -- 3.2. The Harish-Chandra transform -- 3.3. The Mellin transform -- 3.4. The Selberg transform -- 3.5. The principal series of ( ) -- Chapter 4. Maass cusp forms -- 4.1. Cusp forms of weight 0 -- 4.2. Hecke operators -- 4.3. Adelic Maass forms -- Chapter 5. Eisenstein series -- 5.1. Induced representations of ( ) -- 5.2. Definition of Eisenstein series -- 5.3. The finite part of -- 5.4. An orthogonal basis for ( ₁, ₂)^{ _{∞}× ₁( )} -- 5.5. Evaluation of the basis elements -- 5.6. Fourier expansion of Eisenstein series -- 5.7. Meromorphic continuation -- 5.8. Character sums -- Chapter 6. The kernel of ( ) -- 6.1. The spectral decomposition -- 6.2. Kernel functions -- 6.3. A spectral lower bound for _{ℎ*ℎ*}( , ) -- 6.4. The spectral form of the kernel of ( ) -- Chapter 7. A Fourier trace formula for (2) -- 7.1. Convergence of the spectral side -- 7.2. Cuspidal contribution -- 7.3. Residual contribution -- 7.4. Continuous contribution -- 7.5. Geometric side -- 7.6. Final formulas -- 7.7. Classical derivation -- Chapter 8. Validity of the KTF for a broader class of ℎ -- 8.1. Preliminaries -- 8.2. Smooth truncation -- 8.3. Comparing the KTF for ℎ and ℎ_{ } -- 8.4. ₀( ) for not smooth or compactly supported -- 8.5. Proof of Proposition 8.29 -- Chapter 9. Kloosterman sums -- 9.1. A bound for twisted Kloosterman sums -- 9.2. Factorization -- 9.3. Proof of Theorem 9.2 -- Chapter 10. Equidistribution of Hecke eigenvalues -- Bibliography -- Notation index -- Subject index.

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