Indices of nilpotency in certain spaces of modular forms

Abstract

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 $(p, N) \in \{(3, 1), (5, 1), (7, 1), (3, 4)\}$. 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 $p^t$-core partition functions modulo $p$ for $p\in \{3, 5, 7\}$ and $t\geq 1$, and we prove an infinite family of congruences modulo $3$ for the $r$th power partition function, $p_r(n)$, 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.