On the Fourier expansions of Jacobi forms

We use the relationship between Jacobi forms and vector-valued modular forms to study the Fourier expansions of Jacobi forms of indexes p , p 2 , and p q for distinct odd primes p , q . 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 p q or p 2 , there are restrictions on which of the components will uniquely determine the form. Moreover, for indexes of the form p , 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 Sl 2 ( ℤ ) 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.