Preface

On parameters for the group 𝑆𝑂(2𝑛)

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. 𝐿-functions for 𝐺𝐿_{𝑛}×𝐺𝐿_{𝑚} with 𝑚<𝑛

2. Converse Theorems

3. Limiting the rank of the twists

4. Limiting the ramification of the twists

5. Speculation on 𝐺𝐿₁ twists

6. Converse theorems with poles

7. Local converse theorems

8. Final remarks

Acknowledgement

References

A 𝑝-adic integral for the reciprocal of 𝐿-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 𝔭-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 𝐔_{2,2}

1. Introduction

2. Notation

3. \Gln harmonics and invariants on \pvnkl

4. Special highest weight vectors in \calh⊗𝐽

5. The case 𝑛≥4, 𝑝=𝑞=2

6. The case 𝑛=3, 𝑝=𝑞=2

7. The case 𝑛=𝑝=𝑞=2

8. The case 𝑛=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 (𝜒,𝑏)-Theory

8. Endoscopy Correspondence and (𝜏,𝑏)-Theory

9. Final Remarks

References

An inductive formula for 𝜖-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 𝜖-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. 𝑅-groupe et cardinal des paquets stables de représentations tempérées

References

On a certain sum of automorphic 𝐿-functions

1. Automorphic 𝐿-functions as a trace

2. Geometric construction of 𝜓

3. A certain sum of 𝐿-functions

References

Analytic constructions of 𝑝-adic 𝐿-functions and Eisenstein series

1. Complex and 𝑝-adic 𝐿-functions

2. 𝑝-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 𝑝

6. Doubling method and Ikeda’s constructions

Appendix A. Appendix. On 𝑝-adic 𝐿-functions for 𝐺𝑆𝑝(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 𝛾(𝑠,𝜋,Λ²,𝜓_{𝐹})

4. Concluding Remarks

References

CAP forms, Eisenstein series, and some arithmetic applications

1. Introduction

2. Selmer groups

3. Finding extensions

4. Forms on 𝐺𝑆𝑝₄

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 𝐿-functions and automorphic descent

4. On CAP representations of \Sp_{4𝑚}(\BA)

5. On CAP representations of split even orthogonal groups

References

On the singularities of branch curves of 𝐾3 surfaces and applications

1. Introduction

2. Adjoint curves, Branch curves, 𝐾3 surfaces

3. Zariski pairs and 𝐾3 surfaces

References