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El. knyga: Gross-Zagier Formula on Shimura Curves

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  • Formatas: 272 pages
  • Serija: Annals of Mathematics Studies
  • Išleidimo metai: 11-Nov-2012
  • Leidėjas: Princeton University Press
  • Kalba: eng
  • ISBN-13: 9781400845644
Kitos knygos pagal šią temą:
Gross-Zagier Formula on Shimura CurvesDidesnis paveikslėlis
  • Formatas: 272 pages
  • Serija: Annals of Mathematics Studies
  • Išleidimo metai: 11-Nov-2012
  • Leidėjas: Princeton University Press
  • Kalba: eng
  • ISBN-13: 9781400845644
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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.

Preface vii
1 Introduction and Statement of Main Results
1(27)
1.1 Gross-Zagier formula on modular curves
1(1)
1.2 Shimura curves and abelian varieties
2(4)
1.3 CM points and Gross-Zagier formula
6(3)
1.4 Waldspurger formula
9(3)
1.5 Plan of the proof
12(8)
1.6 Notation and terminology
20(8)
2 Weil Representation and Waldspurger Formula
28(30)
2.1 Weil representation
28(8)
2.2 Shimizu lifting
36(4)
2.3 Integral representations of the L-function
40(3)
2.4 Proof of Waldspurger formula
43(1)
2.5 Incoherent Eisenstein series
44(14)
3 Mordell-Weil Groups and Generating Series
58(48)
3.1 Basics on Shimura curves
58(10)
3.2 Abelian varieties parametrized by Shimura curves
68(15)
3.3 Main theorem in terms of projectors
83(8)
3.4 The generating series
91(6)
3.5 Geometric kernel
97(3)
3.6 Analytic kernel and kernel identity
100(6)
4 Trace of the Generating Series
106(65)
4.1 Discrete series at infinite places
106(4)
4.2 Modularity of the generating series
110(7)
4.3 Degree of the generating series
117(5)
4.4 The trace identity
122(6)
4.5 Pull-back formula: compact case
128(10)
4.6 Pull-back formula: non-compact case
138(15)
4.7 Interpretation: non-compact case
153(18)
5 Assumptions on the Schwartz Function
171(13)
5.1 Restating the kernel identity
171(3)
5.2 The assumptions and basic properties
174(4)
5.3 Degenerate Schwartz functions I
178(3)
5.4 Degenerate Schwartz functions II
181(3)
6 Derivative of the Analytic Kernel
184(22)
6.1 Decomposition of the derivative
184(7)
6.2 Non-archimedean components
191(5)
6.3 Archimedean components
196(1)
6.4 Holomorphic projection
197(5)
6.5 Holomorphic kernel function
202(4)
7 Decomposition of the Geometric Kernel
206(24)
7.1 Neron-Tate height
207(9)
7.2 Decomposition of the height series
216(3)
7.3 Vanishing of the contribution of the Hodge classes
219(4)
7.4 The goal of the next chapter
223(7)
8 Local Heights of CM Points
230(21)
8.1 Archimedean case
230(3)
8.2 Supersingular case
233(6)
8.3 Superspecial case
239(5)
8.4 Ordinary case
244(1)
8.5 The j-part
245(6)
Bibliography 251(4)
Index 255
Xinyi Yuan is assistant professor of mathematics at Princeton University. Shou-wu Zhang is professor of mathematics at Princeton University and Columbia University. Wei Zhang is assistant professor of mathematics at Columbia University.
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