The Gross-Zagier Formula on Shimura Curves : : (AMS-184) / / Xinyi Yuan, Wei Zhang, Shou-wu Zhang.
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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Place / Publishing House: | Princeton, NJ : : Princeton University Press, , [2012] ©2013 |
Year of Publication: | 2012 |
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Language: | English |
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Physical Description: | 1 online resource (272 p.) |
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Yuan, Xinyi, author. aut http://id.loc.gov/vocabulary/relators/aut The Gross-Zagier Formula on Shimura Curves : (AMS-184) / Xinyi Yuan, Wei Zhang, Shou-wu Zhang. Course Book Princeton, NJ : Princeton University Press, [2012] ©2013 1 online resource (272 p.) text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda Annals of Mathematics Studies ; 184 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 restricted access http://purl.org/coar/access_right/c_16ec online access with authorization star 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. Issued also in print. Mode of access: Internet via World Wide Web. In English. Description based on online resource; title from PDF title page (publisher's Web site, viewed 31. Jan 2022) Arithmetical algebraic geometry. Automorphic forms. Quaternions. Shimura varieties. MATHEMATICS / Number Theory. bisacsh Arakelov theory. Benedict Gross. Don Zagier. EichlerГhimura theory. Eisenstein series. GrossКagier formula. Heegner point. Hodge bundle. Hodge index theorem. L-series. MordellЗeil group. NeronДate height. RankinГelberg L-function. Schwartz function. Shimizu lifting. Shimura curve. Shimura curves. SiegelЗeil formula. Waldspurger formula. Weil representation. abelian varieties. analytic kernel function. analytic kernel. degenerate Schwartz function. discrete series. generating series. geometric kernel. height series. holomorphic kernel function. holomorphic projection. incoherent Eisenstein series. incoherent automorphic representation. incoherent quaternion algebra. kernel function. kernel identity. local height. modular curve. modularity. multiplicity function. non-archimedean local field. non-degenerate quadratic space. ordinary component. orthogonal space. projector. pull-back formula. ramified quadratic extension. supersingular component. superspecial component. theta function. theta liftings. theta series. trace identity. un-normalized kernel function. unramified quadratic extension. Zhang, Shou-wu, author. aut http://id.loc.gov/vocabulary/relators/aut Zhang, Wei, author. aut http://id.loc.gov/vocabulary/relators/aut Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020 9783110494914 ZDB-23-PMB Title is part of eBook package: De Gruyter Princeton University Press eBook-Package Backlist 2000-2013 9783110442502 print 9780691155920 https://doi.org/10.1515/9781400845644?locatt=mode:legacy https://www.degruyter.com/isbn/9781400845644 Cover https://www.degruyter.com/document/cover/isbn/9781400845644/original |
language |
English |
format |
eBook |
author |
Yuan, Xinyi, Yuan, Xinyi, Zhang, Shou-wu, Zhang, Wei, |
spellingShingle |
Yuan, Xinyi, Yuan, Xinyi, Zhang, Shou-wu, Zhang, Wei, The Gross-Zagier Formula on Shimura Curves : (AMS-184) / Annals of Mathematics Studies ; 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 |
author_facet |
Yuan, Xinyi, Yuan, Xinyi, Zhang, Shou-wu, Zhang, Wei, Zhang, Shou-wu, Zhang, Shou-wu, Zhang, Wei, Zhang, Wei, |
author_variant |
x y xy x y xy s w z swz w z wz |
author_role |
VerfasserIn VerfasserIn VerfasserIn VerfasserIn |
author2 |
Zhang, Shou-wu, Zhang, Shou-wu, Zhang, Wei, Zhang, Wei, |
author2_variant |
s w z swz w z wz |
author2_role |
VerfasserIn VerfasserIn VerfasserIn VerfasserIn |
author_sort |
Yuan, Xinyi, |
title |
The Gross-Zagier Formula on Shimura Curves : (AMS-184) / |
title_sub |
(AMS-184) / |
title_full |
The Gross-Zagier Formula on Shimura Curves : (AMS-184) / Xinyi Yuan, Wei Zhang, Shou-wu Zhang. |
title_fullStr |
The Gross-Zagier Formula on Shimura Curves : (AMS-184) / Xinyi Yuan, Wei Zhang, Shou-wu Zhang. |
title_full_unstemmed |
The Gross-Zagier Formula on Shimura Curves : (AMS-184) / Xinyi Yuan, Wei Zhang, Shou-wu Zhang. |
title_auth |
The Gross-Zagier Formula on Shimura Curves : (AMS-184) / |
title_alt |
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 |
title_new |
The Gross-Zagier Formula on Shimura Curves : |
title_sort |
the gross-zagier formula on shimura curves : (ams-184) / |
series |
Annals of Mathematics Studies ; |
series2 |
Annals of Mathematics Studies ; |
publisher |
Princeton University Press, |
publishDate |
2012 |
physical |
1 online resource (272 p.) Issued also in print. |
edition |
Course Book |
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 |
isbn |
9781400845644 9783110494914 9783110442502 9780691155920 |
callnumber-first |
Q - Science |
callnumber-subject |
QA - Mathematics |
callnumber-label |
QA242 |
callnumber-sort |
QA 3242.5 Y83 42017 |
url |
https://doi.org/10.1515/9781400845644?locatt=mode:legacy https://www.degruyter.com/isbn/9781400845644 https://www.degruyter.com/document/cover/isbn/9781400845644/original |
illustrated |
Not Illustrated |
dewey-hundreds |
500 - Science |
dewey-tens |
510 - Mathematics |
dewey-ones |
516 - Geometry |
dewey-full |
516.352 |
dewey-sort |
3516.352 |
dewey-raw |
516.352 |
dewey-search |
516.352 |
doi_str_mv |
10.1515/9781400845644?locatt=mode:legacy |
oclc_num |
979881796 |
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Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020 Title is part of eBook package: De Gruyter Princeton University Press eBook-Package Backlist 2000-2013 |
is_hierarchy_title |
The Gross-Zagier Formula on Shimura Curves : (AMS-184) / |
container_title |
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