Indices of nilpotency in certain spaces of modular forms

TL;DR

This paper investigates the nilpotency indices of Hecke operators on modular forms modulo primes, establishing bounds and deriving new congruences for partition functions, with conjectures on degree lowering related to the Delta function.

Contribution

It provides new upper bounds on nilpotency indices and derives infinite families of congruences for partition functions in specific modular forms settings.

Findings

Established bounds on nilpotency indices for certain primes and levels.

Proved infinite families of congruences for p^t-core partition functions modulo p.

Proposed conjectures on degree lowering and nilpotency related to the Delta function.

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.