Additive decompositions for rings of modular forms

TL;DR

This paper investigates the structure of rings of integral modular forms, revealing their decompositions, and characterizes conditions for Cohen--Macaulay properties and finite generation using vector bundle decompositions.

Contribution

It provides new decomposition results for modules of modular forms and applies these to establish Cohen--Macaulay conditions and finite generation criteria.

Findings

Rings of modular forms often decompose into well-understood modules

Characterization of Cohen--Macaulay rings among modular forms

Finite generation results for rings of modular forms

Abstract

We study rings of integral modular forms for congruence subgroups as modules over the ring of integral modular forms for the full modular group. In many cases these modules are free or decompose at least into well-understood pieces. We apply this to characterize which rings of modular forms are Cohen--Macaulay and to prove finite generation results. These theorems are based on decomposition results about vector bundles on the compactified moduli stack of elliptic curves.