Indices of nilpotency in certain spaces of modular forms

2024/10/31 by Boylan, Matthew, Swati
#11F11 #11F33 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2410.24182

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) ∈ \(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∈ \3, 5, 7\ and t≥ 1, and we prove an infinite family of congruences modulo 3 for the rth 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.

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