Abstract
A well-known observation of Serre and Tate is that the Hecke algebra acts locally nilpotently on modular forms mod 2 on \({{\,\mathrm{\textrm{SL}}\,}}_2({{\,\mathrm{{\mathbb {Z}}}\,}})\). 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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The SAGE code for Algorithm 1 can be found at the following page: https://alejandrodlpc.github.io/files/hecke-nilpotency.ipynb.
Notes
SAGE code for this algorithm is available at https://alejandrodlpc.github.io/files/hecke-nilpotency.ipynb.
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Acknowledgements
The authors thank Professor Ken Ono for suggesting this problem when we were in his 2022 REU Program at the University of Virginia, and for his continued guidance. They would also like to thank Alejandro De Las Penas Castano, Badri Pandey, and Wei-Lun Tsai for valuable discussions, as well as an anonymous referee for helpful comments.
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Catherine Cossaboom and Sharon Zhou were supported by NSF Grants DMS-2002265, DMS-2055118, DMS-2147273, NSA Grant H98230-22-1-0020, and the Templeton World Charity Foundation.
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Cossaboom, C., Zhou, S. Hecke nilpotency for modular forms mod 2 and an application to partition numbers. Ramanujan J 62, 899–923 (2023). https://doi.org/10.1007/s11139-023-00720-6
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DOI: https://doi.org/10.1007/s11139-023-00720-6