Finite Quadratic Modules and Weil Representations over Number Fields

In the study of Hilbert, Jacobi and orthogonal modular forms
of low weight over number fields it is essential to understand the
representations of Hilbert modular groups or of certain two-fold
central extensions. In the case of the field of natural numbers it is
known that the key to the study of all representations of the modular
group $SL(2,Z)$ which are interesting in the mentioned context are
the Weil representations associated to finite quadratic modules. In
analogy to the case of the field of rational numbers we developed a
theory of finite quadratic modules and their associated Weil
representations over arbitrary number fields. In this talk we report
about the main features of this new theory, about interesting new
phenomena arising in the general theory over arbitrary number fields,
and we indicate applications to the explicit construction of automorphic
forms over number fields.