On finiteness theorems for automorphic forms

TL;DR

This paper establishes finiteness theorems for automorphic forms on Shimura varieties under certain conditions, utilizing sheaf-theoretic methods and extending results to Siegel modular forms.

Contribution

It proves new finiteness theorems for automorphic forms on Shimura data and refines results for Siegel modular forms using advanced sheaf techniques.

Findings

Finiteness theorems for automorphic forms on Shimura varieties.

Application of reflexive sheaves and Koecher principle.

Enhanced finiteness results for Siegel modular forms.

Abstract

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.