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

well-known observation of Serre and Tate is that the Hecke algebra acts locally nilpotently on modular forms mod 2 on SL _2( 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.