The Gross-Zagier formula on Shimura curves / Xinyi Yuan, Shou-wu Zhang, and Wei Zhang.

Format:

Book

Author/Creator:

Yuan, Xinyi, 1981-

Contributor:

Zhang, Shouwu.

Zhang, Wei, 1981-

Series:

Annals of mathematics studies ; no. 184.

Annals of mathematics studies ; no. 184

Language:

English

Subjects (All):

Shimura varieties.

Arithmetical algebraic geometry.

Automorphic forms.

Quaternions.

Physical Description:

vi, 256 pages ; 25 cm.

Place of Publication:

Princeton : Princeton University Press, 2013.

Summary:

Princeton University Press is proud to have published the Annals of Mathematics Studies since 1940. One of the oldest and most respected series in science publishing, it has included many of the most important and influential mathematical works of the twentieth century. The series continues this tradition into the twenty-first century as Princeton University Press publishes the major works of the new millennium. To mark the continued success of the series, all books are again available in paperback. For a complete list of titles, please visit the Princeton University Press web site: www.press.princeton.edu. The most recently published volumes include: Mumford-Tate Groups and Domains: Their Geometry and Arithmetic, Mark Green, Phillip A. Griffiths, and Matt Kerr, The Decomposition of Global Conformal Invariants, Spyros Alexakis, Some Problems of Unlikely Intersections in Arithmetic and Geometry, Umberto Zannier, Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms, Nicholas M. Katz, Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces, Joram, Lindenstrauss, David Preiss, and Jaroslav Tišer, The Ambient Metric, Charles Fefferman and C. Robin Graham, Hypoelliptic Laplacian and Orbital Integrals, Jean-Michel Bismut, Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime, edited by Bas Edixhoven and Jean-Marc Couveignes, Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory, Ben Brubaker, Daniel Bump, and Solomon Friedberg, Introduction to Ramsey Spaces, Stevo Todorcevic, On the Cohomology of Certain Non-Compact Shimura Varieties, Sophie Morel, The Ergodic Theory of Lattice Subgroups, Alexander Gorodnik and Amos Nevo Book jacket.

Contents:

1 Introduction and Statement of Main Results 1

1.1 Gross-Zagier formula on modular curves 1

1.2 Shimura curves and abelian varieties 2

1.3 CM points and Gross-Zagier formula 6

1.4 Waldspurger formula 9

1.5 Plan of the proof 12

1.6 Notation and terminology 20

2 Weil Representation and Waldspurger Formula 28

2.1 Weil representation 28

2.2 Shimizu lifting 36

2.3 Integral representations of the L-function 40

2.4 Proof of Waldspurger formula 43

2.5 Incoherent Eisenstein series 44

3 Mordell-Weil Groups and Generating Series 58

3.1 Basics on Shimura curves 58

3.2 Abelian varieties parametrized by Shimura curves 68

3.3 Main theorem in terms of projectors 83

3.4 The generating series 91

3.5 Geometric kernel 97

3.6 Analytic kernel and kernel identity 100

4 Trace of the Generating Series 106

4.1 Discrete series at infinite places 106

4.2 Modularity of the generating series 110

4.3 Degree of the generating series 117

4.4 The trace identity 122

4.5 Pull-back formula: compact case 128

4.6 Pull-back formula: non-compact case 138

4.7 Interpretation: non-compact case 153

5 Assumptions on the Schwartz Function 171

5.1 Restating the kernel identity 171

5.2 The assumptions and basic properties 174

5.3 Degenerate Schwartz functions I 178

5.4 Degenerate Schwartz functions II 181

6 Derivative of the Analytic Kernel 184

6.1 Decomposition of the derivative 184

6.2 Non-archimedean components 191

6.3 Archimedean components 196

6.4 Holomorphic projection 197

6.5 Holomorphic kernel function 202

7 Decomposition of the Geometric Kernel 206

7.1 Néron-Tate height 207

7.2 Decomposition of the height series 216

7.3 Vanishing of the contribution of the Hodge classes 219

7.4 The goal of the next chapter 223

8 Local Heights of CM Points 230

8.1 Archimedean case 230

8.2 Supersingular case 233

8.3 Superspecial case 239

8.4 Ordinary case 244

8.5 The j-part 245.

Notes:

Includes bibliographical references and index.

ISBN:

9780691155913

0691155917

9780691155920

0691155925

OCLC:

791488485