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The Geometric and Arithmetic Volume of Shimura Varieties of Orthogonal Type

Fritz Hörmann Albert-Ludwigs-Universität Freiburg, Freiburg, Germany
A co-publication of the AMS and Centre de Recherches Mathématiques
Available Formats:
Hardcover ISBN: 978-1-4704-1912-7
Product Code: CRMM/35
List Price: $84.00 MAA Member Price:$75.60
AMS Member Price: $67.20 Electronic ISBN: 978-1-4704-1958-5 Product Code: CRMM/35.E List Price:$79.00
MAA Member Price: $71.10 AMS Member Price:$63.20
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This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $126.00 MAA Member Price:$113.40
AMS Member Price: $100.80 Click above image for expanded view The Geometric and Arithmetic Volume of Shimura Varieties of Orthogonal Type Fritz Hörmann Albert-Ludwigs-Universität Freiburg, Freiburg, Germany A co-publication of the AMS and Centre de Recherches Mathématiques Available Formats:  Hardcover ISBN: 978-1-4704-1912-7 Product Code: CRMM/35  List Price:$84.00 MAA Member Price: $75.60 AMS Member Price:$67.20
 Electronic ISBN: 978-1-4704-1958-5 Product Code: CRMM/35.E
 List Price: $79.00 MAA Member Price:$71.10 AMS Member Price: $63.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$126.00 MAA Member Price: $113.40 AMS Member Price:$100.80
• Book Details

CRM Monograph Series
Volume: 352014; 152 pp
MSC: 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.

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

• Requests

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Volume: 352014; 152 pp
MSC: 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.