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arXiv · 2207.14768

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

Catherine Cossaboom·Sharon Zhou
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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.

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Catherine Cossaboom, Sharon Zhou. 2022-07-29. Hecke nilpotency for modular forms mod 2 and an application to partition numbers. https://arxiv.org/abs/2207.14768

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