Automorphic Forms and Related Geometry : Assessing the Legacy of I.I. Piatetski-Shapiro.

Intro -- Preface -- On parameters for the group (2 ) -- 1. Background -- 2. Statement of the local theorem -- 3. The problem of uniqueness -- 4. The global theorem -- 5. Proof of the local theorem -- References -- Piatetski-Shapiro's Work on Converse Theorems -- 1. -functions for _{ }× _{ } with &lt -- -- 2. Converse Theorems -- 3. Limiting the rank of the twists -- 4. Limiting the ramification of the twists -- 5. Speculation on ₁ twists -- 6. Converse theorems with poles -- 7. Local converse theorems -- 8. Final remarks -- Acknowledgement -- References -- A -adic integral for the reciprocal of -functions -- 1. Introduction and Motivation -- 2. Mazur's measure and its real analytic interpretation -- 3. Fourier coefficients of classical Eisenstein series -- 4. Proof of \thmref{th1} -- Acknowledgements -- References -- Harmonic analysis on symmetric spaces as complex analysis -- 1. Horospherical reduction of complex symmetric spaces (geometrical picture) -- 2. Horospherical reduction of complex symmetric spaces (analytic construction of intertwining operators) -- 3. Dual horospherical Cauchy transform -- 4. Cauchy formula on complex symmetric spaces -- 5. Horospherical duality for real symmetric spaces. Compact spaces -- 6. Horospherical duality for complex crowns of symmetric spaces -- 7. Tube Stein manifolds on causal symmetric spaces -- References -- Testing rationality of coherent cohomology of Shimura varieties -- Introduction -- 1. Restrictions of discrete series representations and global consequences -- 2. Restrictions of minimal types -- 3. Coherent cohomology of unitary group Shimura varieties -- 4. Gross-Prasad periods as cup products -- Acknowledgements -- References -- Hecke fields of Hilbert modular analytic families -- 1. Hilbert Modular Forms of -power level -- 2. Weil numbers.

3. Theorems and conjectures -- 4. Rigidity lemmas -- 5. A Frobenius eigenvalue formula -- 6. Proof of the horizontal theorem: \thmref{Hthm} -- 7. Relative version -- References -- Structure of holomorphic unitary representations: the case of _{2,2} -- 1. Introduction -- 2. Notation -- 3. \Gln harmonics and invariants on \pvnkl -- 4. Special highest weight vectors in \calh⊗ -- 5. The case ≥4, = =2 -- 6. The case =3, = =2 -- 7. The case = = =2 -- 8. The case =1 -- 9. Application to holomorphic unitary representations -- References -- Mellin Transform of Whittaker functions -- 1. Introduction -- 2. Proof of the Theorem -- References -- Automorphic Integral Transforms for Classical Groups I: Endoscopy Correspondences -- 1. Introduction -- 2. Arthur Parametrization of Discrete Spectrum -- 3. Endoscopy Structures for Classical Groups -- 4. Fourier Coefficients and Nilpotent Orbits -- 5. Constructions of the Automorphic Kernel Functions -- 6. Automorphic Descents and Automorphic Forms of Simple Type -- 7. Theta Correspondence and ( , )-Theory -- 8. Endoscopy Correspondence and ( , )-Theory -- 9. Final Remarks -- References -- An inductive formula for -factors -- 1. Introduction -- Notation and Conventions -- 2. Whittaker Models and Functional Equations -- 3. Construction of Whittaker Functions after Paskunas-Stevens -- 4. An inductive formula of -factors -- 5. Proof of Theorem 4.4.3 -- Acknowledgement -- References -- On a new functional equation for local integrals -- 1. Introduction and statement of main result -- 2. Basic estimates -- 3. Model transition -first case -- 4. Model transition -second case -- 5. The functional equation -- Appendix A. Convergence results -- Appendix B. More functional equations -- Acknolwedgement -- References -- Paquets stables des séries discrètes accessibles par endoscopie tordue.

leur paramètre de Langlands -- 1. Introduction -- 2. Notations et propriétés générales -- 3. Le cas des représentations cuspidales -- 4. Support cuspidal étendu et ensemble de blocs de Jordan -- 5. Morphisme dans le L-groupe -- 6. Morphisme de Langlands des paquets stables de séries discrètes -- 7. -groupe et cardinal des paquets stables de représentations tempérées -- References -- On a certain sum of automorphic -functions -- 1. Automorphic -functions as a trace -- 2. Geometric construction of -- 3. A certain sum of -functions -- References -- Analytic constructions of -adic -functions and Eisenstein series -- 1. Complex and -adic -functions -- 2. -adic meromorphic continuation of the Siegel-Eisenstein series -- 3. Pseudomeasures and their Mellin transform -- 4. Application to Minkowski-Siegel Mass constants -- 5. Link to Shahidi's method for SL(2) and regular prime -- 6. Doubling method and Ikeda's constructions -- Appendix A. Appendix. On -adic -functions for (4) -- References -- On stability of root numbers -- Introduction -- 1. What is Stability? -- 2. Towards a general stability -- 3. Stability and equality of factors in the case ( , ,Λ², _{ }) -- 4. Concluding Remarks -- References -- CAP forms, Eisenstein series, and some arithmetic applications -- 1. Introduction -- 2. Selmer groups -- 3. Finding extensions -- 4. Forms on ₄ -- 5. Eisenstein series on unitary groups -- 6. Some comments -- References -- Automorphic descent: an outgrowth from Piatetski-Shapiro's vision -- 1. Introduction -- 2. Rankin-Selberg integrals -- 3. Poles of -functions and automorphic descent -- 4. On CAP representations of \Sp_{4 }(\BA) -- 5. On CAP representations of split even orthogonal groups -- References -- On the singularities of branch curves of 3 surfaces and applications -- 1. Introduction.

2. Adjoint curves, Branch curves, 3 surfaces -- 3. Zariski pairs and 3 surfaces -- References.

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