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An arithmetic Hilbert–Samuel theorem for singular hermitian line bundles and cusp forms

Part of: Differential operators in several variables Arithmetic problems. Diophantine geometry Holomorphic fiber spaces

Published online by Cambridge University Press:  19 August 2014

Robert J. Berman  and
Gerard Freixas i Montplet
Robert J. Berman
Affiliation:
Chalmers Tekniska Högskola and Göteborgs universitet, Göteborg, Sweden email robertb@chalmers.se
Gerard Freixas i Montplet
Affiliation:
CNRS, Institut de Mathématiques de Jussieu, Paris, France email gerard.freixas@imj-prg.fr
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Abstract

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We prove arithmetic Hilbert–Samuel type theorems for semi-positive singular hermitian line bundles of finite height. This includes the log-singular metrics of Burgos–Kramer–Kühn. The results apply in particular to line bundles of modular forms on some non-compact Shimura varieties. As an example, we treat the case of Hilbert modular surfaces, establishing an arithmetic analogue of the classical result expressing the dimensions of spaces of cusp forms in terms of special values of Dedekind zeta functions.

Keywords

Arakelov theoryheightscusp formspluripotential theoryMonge–Ampère operatorsfinite energy functions

MSC classification

Secondary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights 32L05: Holomorphic bundles and generalizations 14G35: Modular and Shimura varieties 32W20: Complex Monge-Ampère operators
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 10 , October 2014 , pp. 1703 - 1728
Copyright
© The Author(s) 2014 

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