Stable Klingen Vectors and Paramodular Newforms [[electronic resource] /] / by Jennifer Johnson-Leung, Brooks Roberts, Ralf Schmidt

• Autore: Johnson-Leung Jennifer
• Pubblicazione: Cham : , : Springer Nature Switzerland : , : Imprint : Springer, , 2023
• Edizione: 1st ed. 2023.
• Descrizione fisica: 1 online resource (XVII, 362 p.)
• Disciplina: 512.7
• Soggetto topico: Number theory; Topological groups; Lie groups; Global analysis (Mathematics); Manifolds (Mathematics); Number Theory; Topological Groups and Lie Groups; Global Analysis and Analysis on Manifolds
• Persona (resp. second.): RobertsBrooks; SchmidtRalf
• Nota di contenuto: 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 -- 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.
• Sommario/riassunto: 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.
• Titolo autorizzato: Stable Klingen Vectors and Paramodular Newforms
• ISBN: 3-031-45177-5
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910799212403321
• Lo trovi qui: Univ. Federico II
• Serie: Lecture Notes in Mathematics, . 1617-9692 ; ; 2342