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Heegner points and Rankin L-series /

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:
Acceso en línea:Texto completo

MARC

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245 0 0 |a Heegner points and Rankin L-series /  |c edited by Henri Darmon, Shou-Wu Zhang. 
260 |a Cambridge, UK ;  |a New York, NY, USA :  |b Cambridge University Press,  |c ©2004. 
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490 1 |a Mathematical Sciences Research Institute publications ;  |v 49 
504 |a Includes bibliographical references and index. 
588 0 |a Print version record. 
520 |a 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 workshop on Special Values of Rankin L-series held at the MSRI in December 2001, is a collection of thirteen articles written by many of the leading contributors in the field, having the Gross-Zagier formula and its avatars as a common unifying theme. It serves as a valuable reference for mathematicians wishing to become further acquainted with the theory of complex multiplication, automorphic forms, the Rankin-Selberg method, arithmetic intersection theory, Iwasawa theory, and other topics related to the Gross-Zagier formula. 
505 0 |a 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. 
505 8 |a 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. 
505 8 |a 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. 
505 8 |a 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. 
505 8 |a 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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700 1 |a Zhang, Shouwu. 
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