The Gross-Zagier Formula on Shimura Curves : : (AMS-184) /

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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Superior document:Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020
VerfasserIn:
Yuan, Xinyi,
Zhang, Shou-wu,
Zhang, Wei,
Place / Publishing House:Princeton, NJ : : Princeton University Press, , [2012]
©2013
Year of Publication:2012
Edition:Course Book
Language:English
Series:Annals of Mathematics Studies ; 184
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Physical Description:1 online resource (272 p.)
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spelling 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 University Press eBook-Package Backlist 2000-2013
is_hierarchy_title The Gross-Zagier Formula on Shimura Curves : (AMS-184) /
container_title Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020
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