Weil representations and Jacobi forms of singular weight over number fields

Abstract

We propose a definition of Jacobi forms over totally real number fields. The under-standing of these Jacobi forms depends very much on the development of a theory ofcertain Weil representations of the Hilbert modular groups. We report on our recentprogress to set up such a theory, on a newly created theory of finite quadratic modulesover number fields, on their associated Weil representations, how to decompose theseand various other results concerning these representations. As an application, we deducefrom our results an explicit description of all Jacobi forms of singular weight over anygiven totally real number field.