Dimension formulae for Jacobi forms

Question

I'm interested in locating dimension formulae for (more general) Jacobi forms associated with a lattice $L$ (where the Jacobi forms of Eichler-Zagier correspond to $L=A_1$).

Unfortunately, the literature seems rather diffuse and I'm rather hoping someone might be able to point me in the direction of some references.

When $L=A_1$, there are formulae due to Eichler-Zagier (and for the scaling $L=A_1(m)$ due to Skoruppa / Skoruppa-Zagier).

However, for general $L$ there seem to be far fewer results. In fact, I'm only aware of the 1992 results of Arakawa, who relates the dimension of spaces of Jacobi cusp forms to the orders of zeros of the Selberg Zeta function. This seems surprising: are there any more?

Answer 1

For general lattices, there is no explicit formula to calculate the dimension of the space of Jacobi forms. For the special lattice $E_8$, an explicit dimensional formula can be found in the paper The theory of Jacobi forms over the Cayley numbers. If you are only interested in a certain lattice, you may consider the isomorphism between spaces of different types of modular forms to seek a dimensional formula. Another way is to consider Jacobi forms as vector-valued modular forms for the Weil representation. Then it is possible to calculate the dimension of the space of Jacobi forms by the dimensional formulas of the space of vector-valued modular forms.