An arithmetic Hilbert-Samuel theorem for singular hermitian line bundles and cusp forms
Number Theory
2019-02-20 v1 Algebraic Geometry
Differential Geometry
Abstract
We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular hermitian line bundles of finite height. In particular, the theorem applies to the log-singular metrics of Burgos-Kramer-K\"uhn. Our theorem is thus suitable for application to some non-compact Shimura varieties with their bundles of cusp forms. As an application, 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.
Cite
@article{arxiv.1209.0996,
title = {An arithmetic Hilbert-Samuel theorem for singular hermitian line bundles and cusp forms},
author = {Robert Berman and Gerard Freixas i Montplet},
journal= {arXiv preprint arXiv:1209.0996},
year = {2019}
}