P-Adic And Combinatorial Properties Of Modular Form Coefficients

For two particular classes of elliptic curves, we establish congruences relating the coefficients of their corresponding modular forms to combinatorial objects. These congruences resemble a supercongruence for the Apery numbers conjectured by Beukers and proved by Ahlgren and Ono in [1]. We also consider the trace Tr-2k(Gamma(0)(N), n) of the Hecke operator T-n acting on the space of cusp forms S-2k(Gamma(0)(N)). We show that for (n, N) = 1, these traces interpolate p-adically in the weight aspect.