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

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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.