p-ADIC AND COMBINATORIAL PROPERTIES OF MODULAR FORM COEFFICIENTS

A well known result is that if E2k is the Eisenstein series of weight 2k and 2k = 2k′ (mod (p− 1)p), then E2k = E2k′ (mod p). In words, this result tells us that the Eisenstein series form a natural family of modular forms that are p-adically interpolated in the weight aspect. Motivated by a question of Serre, we construct a second natural family of modular forms that have this same property. Before proving such a result we will build up the necessary machinery by demonstrate congruences between the coefficients of two infinite families of modular forms and combinatorially defined objects.