Springer, London/Berlin , London, 2015
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
research monograph a theory of Jacobi forms over arbitrary totally real number
fields. However, we concentrate here mainly on the connection of such Jacobi forms
and the theory of Weil representations, leaving out important topics like Hecke
theory and liftings to Hilbert modular forms, which still have to be developed. We
hope to come back to those topics in later publications, but that the present work
stimulates already further interest in this rich new theory. Here, we develop, first
of all, a theory of finite quadratic modules over number fields and their associated
Weil representations. Next we develop in detail the basics of the theory of Jacobi
forms over number fields and the connection to 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 one. We expect
that these singular Jacobi forms play a similar important role in this newly founded
theory of Jacobi forms over number fields as the Weierstrass sigma function does in
the classical theory of Jacobi forms.