We study the index of nilpotency relative to certain Hecke operators in spaces of modular forms with integer weight and level N with integer coefficients modulo primes p for (p, N) ∈ (3, 1), (5, 1), (7, 1), (3, 4). In these settings, we prove upper bounds on certain indices of nilpotency. As an application of our bounds, we prove infinite families of congruences for pt-core partition functions modulo p for p∈3, 5, 7 and t≥1, and we prove an infinite family of congruences modulo 3 for the rth power partition function, pr (n), when r=12k with gcd (k, 6) =1. We also include conjectures on a function which quantifies degree lowering on powers of the Delta-function by the relevant Hecke operators in these settings, and on the index of nilpotency relative to a modification of this degree-lowering function.
Boylan et al. (Fri,) studied this question.