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We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singularhermitian line bundles of finite height. In particular, the theorem applies tothe log-singular metrics of Burgos-Kramer-K\"uhn. Our theorem is thus suitablefor application to some non-compact Shimura varieties with their bundles ofcusp forms. As an application, we treat the case of Hilbert modular surfaces,establishing an arithmetic analogue of the classical result expressing thedimensions of spaces of cusp forms in terms of special values of Dedekind zetafunctions.

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