  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.
