Jacobi forms, finite quadratic modules and Weil representations over number fields

In analogy to the theory of classical Jacobi forms which has proven to
have various important applications ranging from number theory to
physics, we develop in this thesis a theory of Jacobi forms over
arbitrary totally real number fields. For this end we need to
develop, first of all, a theory of finite quadratic modules over
number fields and their associated Weil representations. As a main
application of our theory, we are able to describe explicitly all
singular Jacobi forms over arbitrary totally real number fields whose
indices have rank 1. We expect that these singular Jacobi forms play
a similar important role in this new founded theory of Jacobi forms
over number fields as the Weierstrass sigma function does in the
classical theory of Jacobi forms.