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

Abstract

In analogy to the theory of classical Jacobi forms which has proven to have variousimportant applications ranging from number theory to physics, we develop in thisresearch monograph a theory of Jacobi forms over arbitrary totally real numberfields. However, we concentrate here mainly on the connection of such Jacobi formsand the theory of Weil representations, leaving out important topics like Hecketheory and liftings to Hilbert modular forms, which still have to be developed. Wehope to come back to those topics in later publications, but that the present workstimulates already further interest in this rich new theory. Here, we develop, firstof all, a theory of finite quadratic modules over number fields and their associatedWeil representations. Next we develop in detail the basics of the theory of Jacobiforms over number fields and the connection to Weil representations. As a mainapplication of our theory, we are able to describe explicitly all singular Jacobi formsover arbitrary totally real number fields whose indices have rank one. We expectthat these singular Jacobi forms play a similar important role in this newly foundedtheory of Jacobi forms over number fields as the Weierstrass sigma function does inthe classical theory of Jacobi forms.