Indices of nilpotency in certain spaces of modular forms

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.