Stable Klingen Vectors and Paramodular Newforms / Jennifer Johnson-Leung, Brooks Roberts, and Ralf Schmidt.

Format:

Book

Author/Creator:

Johnson-Leung, Jennifer, author.

Roberts, Brooks, author.

Schmidt, Ralf, author.

Series:

Lecture notes in mathematics & computer science ; Volume 2342.

Lecture Notes in Mathematics Series ; Volume 2342

Language:

English

Subjects (All):

Forms, Modular.

Fourier analysis.

Physical Description:

1 online resource (XVII, 362 p.)

Edition:

First edition.

Place of Publication:

Cham, Switzerland : Springer Nature Switzerland AG, [2023]

Summary:

This book describes a novel approach to the study of Siegel modular forms of degree two with paramodular level. It introduces the family of stable Klingen congruence subgroups of GSp(4) and uses this family to obtain new relations between the Hecke eigenvalues and Fourier coefficients of paramodular newforms, revealing a fundamental dichotomy for paramodular representations. Among other important results, it includes a complete description of the vectors fixed by these congruence subgroups in all irreducible representations of GSp(4) over a nonarchimedean local field. Siegel paramodular forms have connections with the theory of automorphic representations and the Langlands program, Galois representations, the arithmetic of abelian surfaces, and algorithmic number theory. Providing a useful standard source on the subject, the book will be of interest to graduate students and researchers working in the above fields.

Contents:

Intro

Preface

Contents

Glossary of Notations

Roman Symbols

Greek and Other Symbols

1 Introduction

1.1 Dirichlet Characters

A Common Domain

Characters of the Ideles

L-Functions

Strong Multiplicity One

Concluding Remarks

1.2 Modular Forms I

Automorphic Forms on

Distinguished Vectors

Descent to the Upper Half Plane

Modular Forms

Eigenforms

Oldforms

The Correspondence Theorem

An Example

1.3 Modular Forms II

Local Representations

Three Invariants Derived from the Local Newform

Local Factors

Fourier Coefficients

The Classical L-Function

Atkin-Lehner Operators

Summary

1.4 Paramodular Forms I

Automorphic Forms

Siegel Modular Forms

GSp(4)

Local Paramodular New- and Oldforms

Five Types of Cuspidal, Automorphic Representations

Paramodular Forms

1.5 Paramodular Forms II

Four Invariants Derived from the Local Newform

A Difficulty with Paramodular Hecke Operators

The Present Work

Applications and Significance of Paramodular Forms

1.6 Local Results

A Partition

Structure of the Vs(n)

Paramodular Hecke Eigenvalues

Further Local Results

1.7 Results About Siegel Modular Forms

The Main Theorem

Calculations

A Recurrence Relation

1.8 Further Directions

Part I Local Theory

2 Background

2.1 Some Definitions

The Base Field and Matrices

The Symplectic Similitude Group

Characters and Representations

2.2 Representations

Parabolic Induction

Generic Representations

The List of Non-supercuspidal Representations

Saito-Kurokawa Representations

The P3-Quotient

Zeta Integrals.

2.3 The Paramodular Theory

3 Stable Klingen Vectors

3.1 The Stable Klingen Subgroup

3.2 Stable Klingen Vectors

3.3 Paramodularization

3.4 Operators on Stable Klingen Vectors

3.5 Level Raising Operators

3.6 Four Conditions

3.7 Level Lowering Operators

3.8 Stable Hecke Operators

3.9 Commutation Relations

3.10 A Result About Eigenvalues

4 Some Induced Representations

4.1 Double Coset Representatives

4.2 Stable Klingen Vectors in Siegel Induced Representations

4.3 Non-Existence of Certain Vectors

4.4 Characterization of Paramodular Vectors

5 Dimensions

5.1 The Upper Bound

5.2 The Shadow of a Newform

5.3 Zeta Integrals and Diagonal Evaluation

5.4 Dimensions for Some Generic Representations

5.5 Dimensions for Some Non-Generic Representations

5.6 The Table of Dimensions

5.7 Some Consequences

6 Hecke Eigenvalues and Minimal Levels

6.1 At the Minimal Stable Klingen Level

6.2 Non-Generic Paramodular Representations

6.3 At the Minimal Paramodular Level

6.4 An Upper Block Algorithm

7 The Paramodular Subspace

7.1 Calculation of Certain Zeta Integrals

7.2 Generic Representations

7.3 Non-Generic Representations

7.4 Summary Statements

8 Further Results About Generic Representations

8.1 Non-Vanishing on the Diagonal

8.2 The Kernel of a Level Lowering Operator

8.3 The Alternative Model

8.4 A Lower Bound on the Paramodular Level

9 Iwahori-Spherical Representations

9.1 Some Background

An Extended Tits System

Parahoric Subgroups

The Iwahori-Hecke Algebra

Representations

Volumes

9.2 Action of the Iwahori-Hecke Algebra

Projections and Bases

9.3 Stable Hecke Operators and the Iwahori-Hecke Algebra

9.4 Characteristic Polynomials

Part II Siegel Modular Forms

10 Background on Siegel Modular Forms

10.1 Basic Definitions.

The Symplectic Similitude Group

The Siegel Upper Half-Space

Additional Notation

10.2 Modular Forms

Congruence Subgroups

Fourier-Jacobi Expansions

Adelic Automorphic Forms

Paramodular Old- and Newforms

Paramodular Hecke and Atkin-Lehner Operators

11 Operators on Siegel Modular Forms

11.1 Overview

11.2 Level Raising Operators

11.3 A Level Lowering Operator

11.4 Hecke Operators

11.5 Some Relations Between Operators

12 Hecke Eigenvalues and Fourier Coefficients

12.1 Applications

12.2 Another Formulation

12.3 Examples

Indices and Fourier Coefficients

The Data from PSYW

Verification

12.4 Computing Eigenvalues

12.5 A Recurrence Relation

A Tables

Table A.1: Non-Supercuspidal Representations of GSp(4,F)

Table A.2: Non-Paramodular Representations

Table A.3: Stable Klingen Dimensions

Table A.4: Hecke Eigenvalues

Table A.5: Characteristic Polynomials of T0,1s and T1,0s on Vs(1) in Terms of Inducing Data

Table A.6: Characteristic Polynomials of T0,1s and T1,0s on Vs(1) in Terms of Paramodular Eigenvalues

References

Index.

Notes:

Includes bibliographical references and index.

Description based on print version record.

ISBN:

9783031451775

3031451775