On finiteness theorems for automorphic forms

In this paper, for any Shimura datum $(G,\mathcal{D})$ satisfying reasonable conditions so that many interesting cases satisfy, we prove some finiteness theorems for any graded vector space consisting of automorphic forms on $\mathcal{D}$ of some weights over the graded ring of automorphic forms on $X$ with positive parallel weights. We also discuss the integral base ring which we can work on. To realize automorphic forms as global sections on some coherent sheaves on the minimal compactification, we use the notion of reflexive sheaves and higher Koecher principle due to Kai-Wen Lan. Further, we give a more finer version of finiteness results for Siegel modular forms by using only the results of Chai-Faltings.