1. Fourier analysis on finite abelian groups 2. Appendix: spectral theorem for unitary operators 3. Appendix: cancellation lemma There are Fourier expansions on finite abelian groups essentially identical in form to Fourier expansions of periodic functions on the real line. This follows from the spectral theory of unitary operators on finite-dimensional… (More)
1. Zonal spherical harmonics on SL 2 (C) 2. Formula for triple integrals of eigenfunctions 3. Asymptotics of triple integrals 4. Asymptotics of integrals of n-fold products We determine precise asymptotics in spectral parameters for integrals of n-fold products of zonal spherical harmonics on SL 2 (C). In a variety of situations, integrals of products of… (More)
We obtain second integral moments of automorphic L–functions on adele groups GL 2 over arbitrary number fields, by a spectral decomposition using the structure and representation theory of adele groups GL 1 and GL 2. This requires complete reformulation of the notion of Poincaré series, replacing the collection of classical Poincaré series over GL 2 (Q) or… (More)
1. Simultaneous eigenvectors for finite abelian groups 2. Cancellation lemma, orthogonality of distinct characters 3. Representations of finite abelian groups 4. Fourier expansions on finite abelian groups 5. Appendix: spectral theorem for unitary operators 1. Simultaneous eigenvectors for finite abelian groups For a single linear operator T on a complex… (More)
1. Cautionary example 2. Criterion for essential self-adjointness 3. Examples of essentially self-adjoint operators 4. Appendix: Friedrichs' canonical self-adjoint extensions 5. The following has been well understood for 70-120 years, or longer, naturally not in contemporary terminology. The differential operator T = d 2 dx 2 on L 2 [a, b] or L 2 (R) is a… (More)
We establish a spectral identity for moments of Rankin-Selberg L– functions on GL r × GL r−1 over arbitrary number fields, generalizing our previous results for r = 2.
We break the convexity bound in the t–aspect for L–functions attached to cuspforms f for GL 2 (k) over arbitrary number fields k. The argument uses asymptotics with error term with a power saving, for second integral moments over spectral families of twists L(s, f ⊗ χ) by grossencharacters χ, from our previous paper [Di-Ga]. §0. Introduction In many… (More)