We construct and study a natural compactification ${\bar{M}}^r(N)$ of the moduli scheme $M^r(N)$ for rank-r Drinfeld ${\mathbb{F}}_q[T]$-modules with a structure of level $N\in {\mathbb{F}}_q[T]$. Namely, ${\bar{M}}^r(N)=\mathrm{Proj}\mathbf{Eis}(N)$, the projective variety associated with the graded ring $\mathbf{Eis}(N)$ generated by the Eisenstein series of rank r and level N. We use this to define the ring $\mathbf{Mod}(N)$ of all modular forms of rank r and level N. It equals the integral closure of $\mathbf{Eis}(N)$ in their common quotient field ${\tilde{\mathcal{F}}}_r(N)$. Modular forms are characterized as those holomorphic functions on the Drinfeld space ${\mathrm{\Omega }}^r$ with the right transformation behavior under the congruence subgroup $\mathrm{\Gamma }(N)$ of $\mathrm{\Gamma }=\mathrm{GL}(r,{\mathbb{F}}_q[T])$ (“weak modular forms”) which, along with all their conjugates under $\mathrm{\Gamma }/\mathrm{\Gamma }(N)$, are bounded on the natural fundamental domain F for Γ on ${\mathrm{\Omega }}^r$.

我们构建并研究自然压实 ${\overline{米}}^r(N)$ 模数方案的 ${米}^r(N)$对于 rank- r Drinfeld${\mathbb{F}}_q[吨]$-具有层次结构的模块 $N\in {\mathbb{F}}_q[吨]$. 即，${\overline{米}}^r(N)=\mathrm{项目}\mathbf{艾斯}(N)$, 与分级环相关的射影变异 $\mathbf{艾斯}(N)$由秩为r和水平为N的爱森斯坦级数生成。我们用它来定义环$\mathbf{模组}(N)$等级r和等级N的所有模形式。它等于的积分闭包$\mathbf{艾斯}(N)$ 在它们的公商域中 ${\stackrel{～}{\mathcal{F}}}_r(N)$. 模形式被表征为 Drinfeld 空间上的那些全纯函数${\mathrm{\Omega }}^r$ 在同余子群下具有正确的变换行为 $\mathrm{\Gamma }(N)$ 的 $\mathrm{\Gamma }=\mathrm{GL}(r,{\mathbb{F}}_q[吨])$ （“弱模形式”），连同它们在 $\mathrm{\Gamma }/\mathrm{\Gamma }(N)$, 有界于Γ on的自然基域F${\mathrm{\Omega }}^r$.