International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 48, Pages 2583-2594
On the Fourier expansions of Jacobi forms
Department of Mathematics, State University of New York at Brockport, Brockport 14420, NY, USA
Received 19 January 2004
Copyright © 2004 Howard Skogman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We use the relationship between Jacobi forms and vector-valued modular forms to study the Fourier expansions of Jacobi forms of indexes , , and for distinct odd primes , . Specifically, we show that for such indexes, a Jacobi form is uniquely determined by one of the associated components of the vector-valued modular form. However, in the case of indexes of the form or , there are restrictions on which of the components will uniquely determine the form. Moreover, for indexes of the form , this note gives an explicit reconstruction of the entire Jacobi form from a single associated vector-valued modular form component. That is, we show how to start with a single associated vector component and use specific matrices from to find the other components and hence the entire Jacobi form. These results are used to discuss the possible modular forms of half-integral weight associated to the Jacobi form for different subgroups.