AI 中文总结
本文从格的视角引入带水平结构的秩r Drinfeld模空间$\u2112_N^r$，研究其结构、度量与完备化，定义相应Drinfeld模形式并延拓群作用，证明其尖点形式与已有研究结果一致。
AI 中文摘要
本文引入了秩为r≥1且带水平结构的Drinfeld模（等价于带水平结构的秩r格）构成的空间$\u2112_N^r$，研究了其不可约分支以及作用在其上的群作用。在该空间上定义了一种度量，构造了其完备化$\u2190{\u2112_{N}^{r}}$，并将上述群作用延拓到了完备化空间上。证明了该完备化可分解为多个更小的空间$\u2112_N^s$。将Drinfeld模形式定义为$\u2112_{N}^{r}$上的齐次全纯函数，且在完备化$\u2190{\u2112_{N}^{r}}$上连续，并将上述群作用延拓为模形式空间上的作用。最后，将本文定义的模形式与Basson、Breuer和Pink的模形式以及Gekeler的模形式进行了比较，证明尖点形式（即在边界上为零的形式）是一致的。
英文摘要
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.
Comments46 pages. To be published in Transactions of the AMS
