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

Published online by Cambridge University Press:  19 August 2014

Robert J. Berman
Affiliation:
Chalmers Tekniska Högskola and Göteborgs universitet, Göteborg, Sweden email [email protected]
Gerard Freixas i Montplet
Affiliation:
CNRS, Institut de Mathématiques de Jussieu, Paris, France email [email protected]

Abstract

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.

Type
Research Article
Copyright
© The Author(s) 2014 

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