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

2022/07/29 by Cossaboom, Catherine, Zhou, Sharon
#11F11 #11P83 #20C08 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2207.14768

Abstract

A well-known observation of Serre and Tate is that the Hecke algebra acts locally nilpotently on modular forms mod 2 on SL₂(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 → ∞. As an application, we study the parity of the partition function using Hecke nilpotency.

Related

• Indices of nilpotency in certain spaces of modular forms
• Formes modulaires modulo 2 : l'ordre de nilpotence des opérateurs de Hecke
• Formes modulaires modulo 2 : L'ordre de nilpotence des opérateurs de Hecke (version développée)
• Experimental evidence for Maeda's conjecture on modular forms
• Partitions and Hecke images