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 -- 13. Local Gross-Zagier Formula -- 14. Gross-Zagier Formula in Level ND -- 15. Green's Functions of Heegner Points -- 16. Spectral Decomposition -- 17. Lowering Levels -- 18. Continuous Spectrum -- 19. Periods of Eisenstein Series -- Acknowledgments -- References -- Special Cycles and Derivatives of Eisenstein Series -- I. An Attractive Family of Varieties -- 1. Shimura Varieties of Orthogonal Type -- 2. Algebraic Cycles -- 3. Modular Generating Functions -- 4. Connections with Values of Eisenstein Series -- II. Speculations on the Arithmetic Theory -- 5. Integral Models and Cycles -- 6. Connections with Derivatives of Eisenstein Series -- III. Derivatives of L-Series -- 7. Arithmetic Theta Lifts -- 8. Connections with Derivatives of L-Functions -- Appendix: Shimura Curves -- References.

Faltings Heights and the Derivative of Zagier's Eisenstein Series -- 1. The Chowla-Selberg Formula -- 2. Bost's L21 -Arithmetic Divisors and Intersection Theory -- 3. The Main Result -- 4. Construction of the Green's Function xi(m -- v) -- 5. The proof of Theorem 3.2 -- Acknowledgment -- References -- Elliptic Curves and Analogies Between Number Fields and Function Fields -- 1. Introduction -- 2. Review of the Birch and Swinnerton-Dyer Conjecture over Function Fields -- 3. Function Field Analogues of the Gross-Zagier Theorem -- 4. Ranks over Function Fields -- 5. Rank Bounds -- 6. Ranks over Number Fields -- 7. Algebraic Rank Bounds -- 8. Arithmetic and Geometric Bounds I: Cyclotomic Fields -- 9. Arithmetic and Geometric Bounds II: Function Fields over Number Fields -- Acknowledgements -- References -- Heegner Points and Elliptic Curves of Large Rank over Function Fields -- References -- Periods and Points Attached to Quadratic Algebras -- Introduction -- 1. Classical Heegner Points -- 2. Heegner Points and p-adic Integration -- 3. Forms on Tp multiplication H -- 4. Complex Periods and Heegner Points -- 5. p-adic Periods and Stark-Heegner Points -- 6. Heegner Points and Integration on Hp multiplication Hq -- 7. Periods of Hilbert Modular Forms -- References.