Heegner points and Rankin L-series /

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The seminal formula of Gross and Zagier relating heights of Heegner points to derivatives of the associated Rankin L-series has led to many generalisations and extensions in a variety of different directions, spawning a fertile area of study that remains active to this day. This volume, based on a w...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Otros Autores: Darmon, Henri, 1965-, Zhang, Shouwu
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Cambridge, UK ; New York, NY, USA : Cambridge University Press, ©2004.
Colección:Mathematical Sciences Research Institute publications ; 49.
Temas:
Curves, Elliptic.
L-functions.
Courbes elliptiques.
Fonctions L.
MATHEMATICS > Geometry > Algebraic.
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Cover
  • Half-title
  • Series-title
  • Title
  • Copyright
  • Contents
  • Preface
  • Heegner Points: The Beginnings
  • 1. Prologue: The Opportune Arrival of Heegner Points
  • 2. Prehistory
  • 3. Heegner
  • 4. Simplification and Generalisation
  • 5. 1982
  • References
  • Correspondence
  • Gross to Birch: March 1, 1982
  • Birch to Gross: May 6, 1982
  • Gross to Birch: May 14, 1982
  • Birch to Gross: around September 6, 1982
  • Gross to Birch: September 17, 1982
  • Gross to Birch: December 1, 1982
  • Birch to Gross: December 27, 1982.
  • The Gauss Class Number Problem for Imaginary Quadratic Fields
  • 1. Introduction
  • 2. The Deuring-Heilbronn Phenomenon
  • 3. Existence of L-functions of Elliptic Curves with Triple Zeros
  • 4. Solution of the Class Number One Problem
  • Acknowledgment
  • References
  • Heegner Points and Representation Theory
  • 1. Heegner Points on X0(N)
  • 2. Rankin L-Series and a Height Formula
  • 3. Starting from the L-Function
  • 4. Local Representation Theory
  • 5. Unitary Similitudes
  • 6. The L-Group and Its Symplectic Representation
  • 7. Inner Forms
  • 8. Langlands Parameters
  • 9. Local epsilon Factors.
  • 10. Local Linear Forms
  • 11. Local Test Vectors
  • 12. An Explicit Local Formula
  • 13. Ad grave accent lic Groups
  • 14. A Special Case
  • 15. Automorphic Representations
  • 16. When #S Is Even
  • 17. Global Test Vectors
  • 18. An Explicit Global Formula
  • 19. When #S Is Odd
  • 20. Shimura Varieties
  • 21. Nearby Quaternion Algebras
  • 22. The Global Representation
  • 23. The Global Linear Form
  • 24. Global Test Vectors
  • Acknowledgment
  • References
  • Gross-Zagier Revisited
  • 1. Introduction
  • 2. Some Properties of Abelian Schemes and Modular Curves.
  • 3. The Serre-Tate Theorem and the Grothendieck Existence Theorem
  • 4. Computing Naive Intersection Numbers
  • 5. Intersection Formula Via Hom Groups
  • 6. Supersingular Cases with r A(m) = 0
  • 7. Application of a Construction of Serre
  • 8. Intersection Theory Via Meromorphic Tensors
  • 9. Self-Intersection Formula and Application to Global Height Pairings
  • 10. Quaternionic Explications
  • Appendix by W.R. Mann: Elimination of Quaternionic Sums
  • References
  • Special Value Formulae for Rankin L-Functions
  • 1. Introduction
  • 2. Notation and Hypotheses
  • 3. Atkin-Lehner Theory on GL2.
  • 4. Quaternion Algebras and the Jacquet-Langlands Correspondence
  • 5. The Work of Waldspurger
  • 6. Test Vectors: The Work of Gross and Prasad
  • 7. The Work of Gross and Zhang
  • References
  • Gross-Zagier Formula for GL(2), II
  • 1. Introduction and Notation
  • 2. Automorphic Forms
  • 3. Weights and Levels
  • 4. Automorphic L-Series
  • 5. Rankin-Selberg L-Series
  • 6. The Odd Case
  • 7. The Even Case
  • 8. The Idea of Gross and Zagier
  • 9. Calculus on Arithmetic Surfaces
  • 10. Decomposition of Heights
  • 11. Construction of the Kernels
  • 12. Geometric Pairing.

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