On Drinfeld modular forms of higher rank IV: Modular forms with level

We construct and study a natural compactification ${\bar{M}}^r(N)$ of the moduli scheme $M^r(N)$ for rank-$r$ Drinfeld ${\text{F}}_q[T]$-modules with a structure of level $N\in {\text{F}}_q[T]$. Namely, $\overline{M}^r(N) = {\rm Proj},{\bf Eis}(N)$,theprojectivevarietyassociatedwiththegradedring${\bf Eis}(N)$generatedbytheEisensteinseriesofrank$r$andlevel$N$. We use this to define the ring $\mathbf{M}\mathbf{o}\mathbf{d}(N)$ of all modular forms of rank $r$ and level $N$. It equals the integral closure of $\mathbf{E}\mathbf{i}\mathbf{s}(N)$ in their common quotient field ${\widetilde{\text{MF}}}_r(N)$. Modular forms are characterized as those holomorphic functions on the Drinfeld space ${\text{Om}}^r$ with the right transformation behavior under the congruence subgroup $\text{Ga}(N)$ of $\text{Ga}=\mathrm{G}\mathrm{L}(r,{\text{F}}_q[T])$ (“weak modular forms”) which, along with all their conjugates under $\text{Ga}\mathrm{/}\text{Ga}(N)$, are bounded on the natural fundamental domain $\text{BF}$ for $\text{Ga}$ on ${\text{Om}}^r$.