The new theory of Jacobi forms over totally real number fields introduced in this monograph is
expected to give further insight into the arithmetic theory of Hilbert modular forms its
L-series and into elliptic curves over number fields. This work is inspired by the classical
theory of Jacobi forms over the rational numbers which is an indispensable tool in the
arithmetic theory of elliptic modular forms elliptic curves and in many other disciplines in
mathematics and physics. Jacobi forms can be viewed as vector valued modular forms which take
values in so-called Weil representations. Accordingly the first two chapters develop the
theory of finite quadratic modules and associated Weil representations over number fields. This
part might also be interesting for those who are merely interested in the representation theory
of Hilbert modular groups. One of the main applications is the complete classification of
Jacobi forms of singular weight over an arbitrary totally real number field.
