Weil representations and Jacobi forms of singular weight over number fields

25th Automorphic Forms Workshop, Oregon, Amerika Birleşik Devletleri, 23 - 26 Mart 2011, ss.4

We propose a definition of Jacobi forms over totally real number fields. The under-

standing of these Jacobi forms depends very much on the development of a theory of

certain Weil representations of the Hilbert modular groups. We report on our recent

progress to set up such a theory, on a newly created theory of finite quadratic modules

over number fields, on their associated Weil representations, how to decompose these

and various other results concerning these representations. As an application, we deduce

from our results an explicit description of all Jacobi forms of singular weight over any

given totally real number field.