Non-Ordinary Primes: A Story

This research was supported in part by grant DMS from the National Science Foundation A normalized modular eigenform f is said to be ordinary at a prime p if p does not divide the p-th Fourier coefficient of f. We take f to be a modular form of level 1 and weight k f12, 16, 18, 20, 22, 26g and search for primes where f is not ordinary. To do this, we need an efficient way to compute the reduction modulo p of the p-th Fourier coefficient. A convenient formula was known for k = 12; trying to understand it leads to generalized Rankin–Cohen brackets and thence to formulas that we can use to look for non-ordinary primes. We do this for p 1 000 000.