Additive decompositions for rings of modular forms

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.