Martin Eichler

The Ring of Jacobi Forms

The object of this and the following section is to obtain as much information as possible about the algebraic structure of the set of Jacobi forms, in particular about i) the dimension of Jk,m (k,m fixed), i.e. the structure of this space as a vector space over ℂ; ii) the additive structure of \( {J_{*,m}} = \mathop \oplus \limits_k {J_{k,m}} \) (m fixed) as a module over the graded ring \( {M_*} = \mathop \oplus \limits_k {M_k} \) of ordinary modular forms; iii) the multiplicative structure of the bigraded ring \( {J_{*,*}} = \mathop \oplus \limits_{k,m} {J_{k,m}} \) of all Jacobi forms. We will study only the case of forms on the full Jacobi group \( \Gamma _1^J \) (and usually only the case of forms of even weight), but many of the considerations could be extended to arbitrary Γ.

Relations with Other Types of Modular Forms

In §2 we showed that the coefficients c(n,r) of a Jacobi form of index m depend only on the “discriminant” r2-4nm and on the value of r(mod 2m), i.e.