An eta-quotient of level N is a modular form of the shape f (z) = δ|N η(δz) rδ . We study the problem of determining levels N for which the graded ring of holomorphic modular forms for Γ 0 (N ) is generated by (holomorphic, respectively weakly holomorphic) eta-quotients of level N . In addition, we prove that if f (z) is a holomorphic modular form that is nonvanishing on the upper half plane and has integer Fourier coefficients at infinity, then f (z) is an integer multiple of an eta-quotient. Finally, we use our results to determine the structure of the cuspidal subgroup of J 0 (2 k )(Q).

Let b 13 (n) denote the number of 13-regular partitions of n. We study in this paper the behavior of b 13 (n) modulo 3 where n ≡ 1 (mod 3). In particular, we identify an infinite family of arithmetic progressions modulo arbitrary powers of 3 such that b 13 (n) ≡ 0 (mod 3).

Let ≥ 5 be prime, let m ≥ 1 be an integer, and let p(n) denote the partition function. Folsom, Kent, and Ono recently proved that there exists a positive integer b (m) of size roughly m 2 such that the module formed from the Z/ m Z-span of generating functions for p b n+1 24 with odd b ≥ b (m) has finite rank. The same result holds with "odd" b replaced by "even" b. Furthermore, they proved an upper bound on the ranks of these modules. This upper bound is independent of m; it is +12 24. In this paper, we prove, with a mild condition on , that b (m) ≤ 2m − 1. Our bound is sharp in all computed cases with ≥ 29. To deduce it, we prove structure theorems for the relevant Z/ m Z-modules of modular forms. This work sheds further light on a question of Mazur posed to Folsom, Kent, and Ono.