Screen LibraryThe gross-zagier formula on shimura curves /
This comprehensive account of the Gross-Zagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of L-series. The formula will have new applications for the Birch and Swinnerton-Dyer conjecture and Diophantine equations....
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| Main Authors: | |
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| Corporate Authors: | |
| Group Author: | ; |
| Published: |
Princeton University Press,
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| Publisher Address: | Princeton, N.J. : |
| Publication Dates: |
[2013] ©2013 |
| Literature type: | eBook |
| Language: | English |
| Edition: | Course Book. |
| Series: |
Annals of mathematics studies
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| Subjects: | |
| Online Access: |
http://dx.doi.org/10.1515/9781400845644 http://www.degruyter.com/doc/cover/9781400845644.jpg |
| Summary: |
This comprehensive account of the Gross-Zagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of L-series. The formula will have new applications for the Birch and Swinnerton-Dyer conjecture and Diophantine equations. The book begins with a conceptual formulation of the Gross-Zagier formula in terms of incoherent quaternion algebras and incoherent automorphic representations with rational coefficients attached naturally to abelian varieties parametrized by Shimura curves. This is followed by a complete proof of its coherent analogue: the Waldspurger formula, which relates the periods of integrals and the special values of L-series by means of Weil representations. The Gross-Zagier formula is then reformulated in terms of incoherent Weil representations and Kudla's generating series. Using Arakelov theory and the modularity of Kudla's generating series, the proof of the Gross-Zagier formula is reduced to local formulas. The Gross-Zagier Formula on Shimura Curves will be of great use to students wishing to enter this area and to those already working in it. |
| Carrier Form: | 1 online resource (272 pages) : illustrations. |
| ISBN: | 9781400845644 |
| Index Number: | QA242 |
| CLC: | O187 |
| Contents: |
Frontmatter -- Contents -- Preface -- Chapter One. Introduction and Statement of Main Results -- Chapter Two. Weil Representation and Waldspurger Formula -- Chapter Three. Mordell-Weil Groups and Generating Series -- Chapter Four. Trace of the Generating Series -- Chapter Five. Assumptions on the Schwartz Function -- Chapter Six. Derivative of the Analytic Kernel -- Chapter Seven. Decomposition of the Geometric Kernel -- Chapter Eight. Local Heights of CM Points -- Bibliography -- Index. |