| Description |
1 online resource (xiii, 367 pages). |
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data file |
| Physical Medium |
polychrome |
| Series |
Mathematical Sciences Research Institute publications ; 49
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Mathematical Sciences Research Institute publications ; 49.
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| Bibliography |
Includes bibliographical references and index. |
| Summary |
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. |
| Contents |
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. |
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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. |
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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. |
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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. |
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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. |
| Local Note |
eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - North America |
| Subject |
Curves, Elliptic.
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Curves, Elliptic. |
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L-functions.
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L-functions. |
| Genre/Form |
Electronic books.
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Electronic books.
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| Added Author |
Darmon, Henri, 1965-
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Zhang, Shouwu.
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| Other Form: |
Print version: Heegner points and Rankin L-series. Cambridge, UK ; New York, NY, USA : Cambridge University Press, ©2004 (DLC) 2004045108 |
| ISBN |
0511211899 |
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9780511211898 |
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052183659X |
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9780521836593 |
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0511217269 (electronic book) |
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9780511217265 (electronic book) |
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0511215479 (electronic book) |
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9780511215476 (electronic book) |
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9780511756375 (ebook) |
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0511756372 (ebook) |
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