Hardcover ISBN: | 978-1-4704-1912-7 |
Product Code: | CRMM/35 |
List Price: | $130.00 |
MAA Member Price: | $117.00 |
AMS Member Price: | $104.00 |
eBook ISBN: | 978-1-4704-1958-5 |
Product Code: | CRMM/35.E |
List Price: | $125.00 |
MAA Member Price: | $112.50 |
AMS Member Price: | $100.00 |
Hardcover ISBN: | 978-1-4704-1912-7 |
eBook: ISBN: | 978-1-4704-1958-5 |
Product Code: | CRMM/35.B |
List Price: | $255.00 $192.50 |
MAA Member Price: | $229.50 $173.25 |
AMS Member Price: | $204.00 $154.00 |
Hardcover ISBN: | 978-1-4704-1912-7 |
Product Code: | CRMM/35 |
List Price: | $130.00 |
MAA Member Price: | $117.00 |
AMS Member Price: | $104.00 |
eBook ISBN: | 978-1-4704-1958-5 |
Product Code: | CRMM/35.E |
List Price: | $125.00 |
MAA Member Price: | $112.50 |
AMS Member Price: | $100.00 |
Hardcover ISBN: | 978-1-4704-1912-7 |
eBook ISBN: | 978-1-4704-1958-5 |
Product Code: | CRMM/35.B |
List Price: | $255.00 $192.50 |
MAA Member Price: | $229.50 $173.25 |
AMS Member Price: | $204.00 $154.00 |
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Book DetailsCRM Monograph SeriesVolume: 35; 2014; 152 ppMSC: Primary 11; 14
This book outlines a functorial theory of integral models of (mixed) Shimura varieties and of their toroidal compactifications, for odd primes of good reduction. This is the integral version, developed in the author's thesis, of the theory invented by Deligne and Pink in the rational case. In addition, the author develops a theory of arithmetic Chern classes of integral automorphic vector bundles with singular metrics using the work of Burgos, Kramer and Kühn.
The main application is calculating arithmetic volumes or “heights” of Shimura varieties of orthogonal type using Borcherds' famous modular forms with their striking product formula—an idea due to Bruinier–Burgos–Kühn and Kudla. This should be seen as an Arakelov analogue of the classical calculation of volumes of orthogonal locally symmetric spaces by Siegel and Weil. In the latter theory, the volumes are related to special values of (normalized) Siegel Eisenstein series.
In this book, it is proved that the Arakelov analogues are related to special derivatives of such Eisenstein series. This result gives substantial evidence in the direction of Kudla's conjectures in arbitrary dimensions. The validity of the full set of conjectures of Kudla, in turn, would give a conceptual proof and far-reaching generalizations of the work of Gross and Zagier on the Birch and Swinnerton-Dyer conjecture.
Titles in this series are co-published with the Centre de recherches mathématiques.
ReadershipResearch mathematicians and graduate students interested in Shimura varieties, Siegel-Weil theory, Borcherds products, Kudla's conjectures, and Arakelov theory.
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Table of Contents
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Chapters
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Overview
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Integral models of toroidal compactifications of mixed Shimura varieties
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Volumes of orthogonal Shimura varieties
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Additional Material
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RequestsReview Copy – for publishers of book reviewsAccessibility – to request an alternate format of an AMS title
- Book Details
- Table of Contents
- Additional Material
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This book outlines a functorial theory of integral models of (mixed) Shimura varieties and of their toroidal compactifications, for odd primes of good reduction. This is the integral version, developed in the author's thesis, of the theory invented by Deligne and Pink in the rational case. In addition, the author develops a theory of arithmetic Chern classes of integral automorphic vector bundles with singular metrics using the work of Burgos, Kramer and Kühn.
The main application is calculating arithmetic volumes or “heights” of Shimura varieties of orthogonal type using Borcherds' famous modular forms with their striking product formula—an idea due to Bruinier–Burgos–Kühn and Kudla. This should be seen as an Arakelov analogue of the classical calculation of volumes of orthogonal locally symmetric spaces by Siegel and Weil. In the latter theory, the volumes are related to special values of (normalized) Siegel Eisenstein series.
In this book, it is proved that the Arakelov analogues are related to special derivatives of such Eisenstein series. This result gives substantial evidence in the direction of Kudla's conjectures in arbitrary dimensions. The validity of the full set of conjectures of Kudla, in turn, would give a conceptual proof and far-reaching generalizations of the work of Gross and Zagier on the Birch and Swinnerton-Dyer conjecture.
Titles in this series are co-published with the Centre de recherches mathématiques.
Research mathematicians and graduate students interested in Shimura varieties, Siegel-Weil theory, Borcherds products, Kudla's conjectures, and Arakelov theory.
-
Chapters
-
Overview
-
Integral models of toroidal compactifications of mixed Shimura varieties
-
Volumes of orthogonal Shimura varieties