Drinfeld modular forms of higher rank from a lattice-oriented point of view

Abstract

A space $\mathcal{L}_N^r$ of Drinfeld modules of rank $r \geq 1$ with level structure, or equivalently lattices of rank $r$ with level structure, is introduced, and its irreducible components and group actions on it are investigated. A metric is defined on this space, its completion $\overleftarrow{\mathcal{L}_{N}^{r}}$ is established and the aforementioned group actions are extended to the completion. A decomposition of the completion into multiple smaller spaces $\mathcal{L}_N^s$ is proven. Drinfeld modular forms are defined as homogeneous holomorphic functions on $\mathcal{L}_{N}^{r}$ which are continuous on the completion $\overleftarrow{\mathcal{L}_{N}^{r}}$, and the group actions above are extended to actions on the spaces of modular forms. Finally, the modular forms defined here are compared with those of Basson, Breuer, and Pink and with those of Gekeler, and it is shown that the cusp forms (those which are zero on the boundary) coincide.