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

TL;DR

This paper establishes an arithmetic Hilbert-Samuel theorem for singular hermitian line bundles, enabling new insights into the dimensions of cusp form spaces on non-compact Shimura varieties, especially Hilbert modular surfaces.

Contribution

It introduces an arithmetic Hilbert-Samuel theorem applicable to semi-positive singular hermitian line bundles with finite height, including log-singular metrics, and applies it to Hilbert modular surfaces.

Findings

Derived an arithmetic analogue of classical cusp form dimension formulas.

Applied the theorem to non-compact Shimura varieties with cusp form bundles.

Connected dimensions of cusp form spaces to special values of Dedekind zeta functions.

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.