Stable Klingen Vectors and Paramodular Newforms

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
		L-Functions
		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
		Eigenforms
		The Correspondence Theorem
		An Example
	1.5 Paramodular Forms II
		Local Representations
		Four Invariants Derived from the Local Newform
		Local Factors
		L-Functions
		Fourier Coefficients
		The Classical L-Function
		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
		Fourier Coefficients
		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
			Siegel Modular Forms
			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