In [Rus14b] and [Rus14a], we study graded rings of modular forms over congruence subgroups, with coefficients in subrings A of C, and determine bounds of the weights of modular forms constituting a minimal set of generators, as well as on the degree of the generators of the ideal of relations between them. We give an algorithm that computes the structures of these rings, and formulate conjectures on the minimal generating weight for modular forms with coefficients in Z.

We discuss questions of finiteness of systems of Hecke eigenvalues modulo p^m, for a prime p and an integer m ≥ 2, in analogy to the classical theory that already exists for m = 1. In joint work with Ian Kiming and Gabor Wiese ([KRW14]), we show that these questions are intimately related to a question of Buzzard regarding the boundedness of the eld of denition of Hecke eigenforms (over Q_p), and we formulate precise conjectures. We prove the existence of bounds on the weight ltrations of eigenforms modulo p^m, which gives evidence as to the truth of these conjectures. These bounds are made explicit in the case N = 1, p = 2.