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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
The Geometric and Arithmetic Volume of Shimura Varieties of Orthogonal Type
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
The Geometric and Arithmetic Volume of Shimura Varieties of Orthogonal Type
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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
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
  • 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.

    Titles in this series are co-published with the Centre de recherches mathématiques.

    Readership

    Research mathematicians and graduate students interested in Shimura varieties, Siegel-Weil theory, Borcherds products, Kudla's conjectures, and Arakelov theory.

  • Table of Contents
     
     
    • Chapters
    • Overview
    • Integral models of toroidal compactifications of mixed Shimura varieties
    • Volumes of orthogonal Shimura varieties
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Accessibility – to request an alternate format of an AMS title
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.

Titles in this series are co-published with the Centre de recherches mathématiques.

Readership

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
Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
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