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


Boylan H.

Springer, London/Berlin , London, 2015

  • Basım Tarihi: 2015
  • Yayınevi: Springer, London/Berlin 
  • Basıldığı Şehir: London

Özet

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.