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

作者：Berman Robert J^*; Montplet Gerard Freixas i

来源：Compositio Mathematica, 2014, 150(10): 1703-1728.

DOI：10.1112/S0010437X14007325

• Arakelov theory
• heights
• cusp forms
• pluripotential theory
• Monge-Ampere operators
• finite energy functions

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-Kuhn. 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.

• 出版日期2014-10