Cover

Title page

Contents

Chapter 1. Preface

Local transfer and reducibility of induced representations of 𝑝-adic groups of classical type

1. Introduction

2. Local representations and 𝐿-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 𝐿-functions for 𝑈_{2𝑘}×𝑅_{𝐸/𝐹}𝐺𝐿_{𝑚}, (𝑘<𝑚)

1. Introduction

2. The global integral

3. Convergence of the local integrals

4. Non-vanishing of the local integrals at a given point 𝑠₀.

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 𝐿-function

1. Introduction

2. Notation and conventions

3. Instances of algebraicity

4. The Whittaker \regulator s

5. A cohomological interpretation of \Res_{𝑠=1}𝐿(𝑠,Π×Π^{∨})

6. A cohomological interpretation of \Res_{𝑠=1}𝐿(𝑠,Π,\As^{(-1)ⁿ⁻¹})

7. A relation between the bottom Whittaker \regulator and 𝐿(1,Π,\As^{(-1)ⁿ})

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 𝐾_{ℂ}∖𝔾_{ℂ}/𝔹_{ℂ} for classical groups

0. Introduction

1. Diagonal Subgroups

2. Bilinear forms

3. Direct sum decomposition (𝐺𝐿_{𝑝+𝑞}(ℂ),𝔾𝕃_{𝕡}(ℂ)×𝔾𝕃_{𝕢}(ℂ))

4. Orthogonal direct sums ((𝑂_{𝑝+𝑞}(ℂ),𝕆_{𝕡}(ℂ)×𝕆_{𝕢}(ℂ)) and (𝕊𝕡_{2(𝕡+𝕢)}(ℂ),𝕊𝕡_{2𝕡}(ℂ)×𝕊𝕡_{2𝕢}(ℂ))

5. Polarizations ((𝑆𝑝_{2𝑚}(ℂ),𝔾𝕃_{𝕞}(ℂ)) and (𝕆_{2ℓ}(ℂ),𝔾𝕃_{ℓ}(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 𝐺𝐿_{𝑁}

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 𝐺𝑆𝑝₄

3. Vector-valued real analytic Siegel modular forms

4. Infinity type of the associated automorphic representation of 𝐺𝑆𝑝₄

5. Correspondence between automorphic representations of 𝐺𝑆𝑝₄ and 𝐺𝐿₄

6. Application of Rankin-Selberg method

7. Conjecture on the existence of mod ℓ Galois representations

8. Bounds of certain subgroups of 𝐺𝑆𝑝₄(\F_{ℓⁿ})

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 𝑆𝑂(𝑝,𝑞),𝑆𝑈(𝑝,𝑞),𝑆𝑝(𝑝,𝑞)

3. The case of the classical groups 𝑆𝐿_{𝑛}(\R) and 𝑆𝐿_{𝑛}(\C)

References

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