The Geometric and Arithmetic Volume of Shimura Varieties of Orthogonal Type

Fritz Hormann

ISBN 10: 1470419122 / ISBN 13: 9781470419127

Editore: American Mathematical Society, US, 2015

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 Kuhn.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-Kuhn 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. Codice articolo LU-9781470419127

Riassunto:

Hörmann provides mathematicians, students, and researchers with the outline for a functional theory of integral models of mixed Shimura varieties and their toroidal compactifications, for odd primes of good reduction. Hormann expands his theoretical work based on theory invented by Dligne and Pink. Also included is a developed theory of arithmetic Chern classes of integral automorphic vector bundles with singular metrics, based on work by Burgos, Kramer, and Kühn, and material devoted to investigating volumes of orthogonal Shimura varieties. Hörmann is a post-doctoral scholar at Albert Ludwigs University, Freiburg, Germany. Annotation ©2015 Ringgold, Inc., Portland, OR (protoview.com)

Dati bibliografici

Titolo: The Geometric and Arithmetic Volume of ...
Casa editrice: American Mathematical Society, US
Data di pubblicazione: 2015
Legatura: Hardback
Condizione: New