Hecke nilpotency for modular forms mod 2 and an application to partition numbers

TL;DR

This paper investigates the nilpotent action of Hecke operators on modular forms mod 2, providing algorithms and formulas to understand their behavior and applying these insights to the parity of partition numbers.

Contribution

It introduces an algorithm to compute Hecke nilpotency degrees for cusp forms mod 2 and derives a formula for counting such forms of a given nilpotency degree.

Findings

Hecke nilpotency degrees have no limiting distribution as weight increases.

An explicit formula for counting cusp forms mod 2 by nilpotency degree.

Application to the parity of the partition function using Hecke nilpotency.

Abstract

A well-known observation of Serre and Tate is that the Hecke algebra acts locally nilpotently on modular forms mod 2 on $\mathrm{SL}_2(\mathbb{Z})$. We give an algorithm for calculating the degree of Hecke nilpotency for cusp forms, and we obtain a formula for the total number of cusp forms mod 2 of any given degree of nilpotency. Using these results, we find that the degrees of Hecke nilpotency in spaces $M_k$ have no limiting distribution as $k \rightarrow \infty$. As an application, we study the parity of the partition function using Hecke nilpotency.