Advances in the Theory of Automorphic Forms and Their
- 1st ed.
- 1 online resource (386 pages)
- Contemporary Mathematics ; v.664 .
- Contemporary Mathematics .
Cover -- Title page -- Contents -- Chapter 1. Preface -- Local transfer and reducibility of induced representations of -adic groups of classical type -- 1. Introduction -- 2. Local representations and -functions -- 3. Generic local transfers - supercuspidal case -- 4. Reducibility of local representations -- References -- Shintani relation for base change: unitary and elliptic representations -- 1. Introduction -- 2. Notation and basic facts (local) -- 3. Results (local) -- 4. Notation and basic facts (global) -- 5. Results (global) -- 6. Proofs -- 7. Appendix: multiplicity one irreducible subquotients -- References -- On -functions for _× _ _, ( < -- ) -- 1. Introduction -- 2. The global integral -- 3. Convergence of the local integrals -- 4. Non-vanishing of the local integrals at a given point ₀. -- 5. Local gamma factors -- 6. Computation of the local integrals with unramified data -- References -- On the Howe duality conjecture in classical theta correspondence -- 1. Introduction -- 2. Special Case of Theorem 1.2 -- 3. Proof of Theorem 1.2 -- 4. Proof of Theorem 1.3 -- 5. Proof of Proposition 3.1 -- References -- Whittaker rational structures and special values of the Asai -function -- 1. Introduction -- 2. Notation and conventions -- 3. Instances of algebraicity -- 4. The Whittaker \regulator s -- 5. A cohomological interpretation of \Res_ ( ,Π×Π^) -- 6. A cohomological interpretation of \Res_ ( ,Π,\As^) -- 7. A relation between the bottom Whittaker \regulator and (1,Π,\As^) -- References -- Character sums of composite moduli and hybrid subconvexity -- 1. Introduction and main results -- 2. Preliminaries -- 3. Proof of Theorem 1 and Corollary 1 -- References -- A linear algebra description of _∖ _/ _ for classical groups -- 0. Introduction -- 1. Diagonal Subgroups. 2. Bilinear forms -- 3. Direct sum decomposition ( _(ℂ), _(ℂ)× _(ℂ)) -- 4. Orthogonal direct sums (( _(ℂ), _(ℂ)× _(ℂ)) and ( _(ℂ), _(ℂ)× _(ℂ)) -- 5. Polarizations (( _(ℂ), _(ℂ)) and ( _(ℂ), _(ℂ))) -- 6. A more unified viewpoint -- 7. Orbit closures and twisted involutions -- References -- Germs for Kloosterman integrals, a review -- 1. The result -- 2. The first step -- 3. Proof of the Theorem -- References -- Fourier coefficients for automorphic forms on quasisplit classical groups -- 1. Introduction -- 2. Arthur Parameters and the Discrete Spectrum -- 3. A Conjecture on Fourier Coefficients -- 4. Progress towards Conjecture 3.2 -- 5. Other Topics Related to Fourier Coefficients -- 6. Roots exchange and Fourier coefficients -- References -- A generalized Casselman-Shalika formula on _ -- 1. Introduction -- 2. Reviews and Set up -- 3. Hecke algebras and their representations -- 4. Intertwining Operators and Functional Equations -- References -- A conditional construction of Artin representations for real analytic Siegel cusp forms of weight (2,1) -- 1. Introduction -- 2. Preliminaries on ₄ -- 3. Vector-valued real analytic Siegel modular forms -- 4. Infinity type of the associated automorphic representation of ₄ -- 5. Correspondence between automorphic representations of ₄ and ₄ -- 6. Application of Rankin-Selberg method -- 7. Conjecture on the existence of mod ℓ Galois representations -- 8. Bounds of certain subgroups of ₄(\F_) -- 9. Proof of the Main Theorem -- 10. Symmetric cube of elliptic cusp forms of weight 1 -- 11. Siegel cusp forms of solvable type -- References -- Another product for a Borcherds form -- 1. Complex coordinates and lattices -- 2. Theta series and the Borcherds lift -- 3. Fourier-Jacobi expansions. 4. A computation of the regularized integral -- 5. Examples -- 6. Comparison -- References -- On Whittaker-Fourier coefficients of automorphic forms on unitary groups: reduction to a local identity -- 1. Introduction -- 2. Notation and preliminaries -- 3. Representations of unitary type -- Part 1. The case \une -- 4. Fourier-Jacobi coefficients and descent -- 5. Reduction to a local conjecture -- 6. A heuristic argument: case of \U₂⁻ -- Part 2. The case \uno -- 7. Gelfand-Graev coefficients and descent -- 8. Reduction to a local conjecture -- 9. A heuristic argument: case of \U₃⁺ -- References -- Introduction to plectic cohomology -- 1. Introduction -- 2. Analytic cohomology of compact pure Shimura varieties -- 3. Interlude: induction and tensor induction -- 4. Etale cohomology of quaternionic Shimura varieties -- 5. Plectic reflex Galois group -- 6. The ℓ-adic plectic conjecture for pure Shimura varieties -- 7. Plectic ℓ-adic cohomology: the pure case -- 8. The mixed case -- 9. Motivation -- 10. Theta functions and classical zeta elements -- 11. Theta functions and cohomology -- 12. Towards zeta elements over totally real fields -- 13. Plectic theta elements -- 14. Specialisations of plectic Siegel classes and Stark's conjectures -- 15. Plectic constructions involving plectic Siegel classes -- 16. Plectic Hodge theory -- 17. An arithmetic application -- 18. Final speculations -- References -- A comparison of automorphic and Artin L-series of GL(2)-type agreeing at degree one primes -- Introduction -- 1. Notations and preliminaries -- 2. Temperedness -- 3. Entireness -- 4. Descent -- References -- Topologies of nodal sets of random band limited functions -- 1. Introduction -- 2. Outline of proofs -- References -- Geometric cycles, classical groups and related cohomology classes for arithmetic groups -- Introduction. 1. Geometric construction of cohomology classes -- 2. The case of the classical groups ( , ), ( , ), ( , ) -- 3. The case of the classical groups _(\R) and _(\C) -- References -- Back Cover.
This volume contains the proceedings of the workshop on "Advances in the Theory of Automorphic Forms and Their L-functions" held in honor of James Cogdell's 60th birthday, held from October 16-25, 2013, at the Erwin Schrödinger Institute (ESI) at the University of Vienna. The workshop and the papers contributed to this volume circle around such topics as the theory of automorphic forms and their L-functions, geometry and number theory, covering some of the recent approaches and advances to these subjects. Specifically, the papers cover aspects of representation theory of p-adic groups, classification of automorphic representations through their Fourier coefficients and their liftings, L-functions for classical groups, special values of L-functions, Howe duality, subconvexity for L-functions, Kloosterman integrals, arithmetic geometry and cohomology of arithmetic groups, and other important problems on L-functions, nodal sets and geometry.