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.

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