Indices of nilpotency in certain spaces of modular forms
Matthew Boylan, Swati
TL;DR
The paper investigates the index of nilpotency $N_T(f,p)$ for Hecke operators acting on mod-$p$ reductions of spaces of modular forms, establishing explicit linear upper bounds for $N_ heta( ext{Delta}^k,p)$ when $(p,N,f) ext{ are } igig\{(5,1, ext{Delta}),(7,1, ext{Delta}),(3,4,D_2)igig iched$ and a similar bound for $N_ heta(D_2^k,3)$ under $ ext{gcd}(k,6)=1$. It further introduces degreedown functions $D_ ho(z)$ and related conjectures to model degree-lowering behavior, supplemented by computations modulo $5,7,3$ and explicit conjectures on their growth. The main arithmetic consequences are infinite families of congruences for $p^t$-core partition functions modulo $p$ and for the $r$-th power partition function modulo $3$, derived from the nilpotency bounds. Overall, the work advances understanding of local nilpotency in modular forms modulo primes and yields actionable congruence results for classical partition-related arithmetic functions.
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.