Non-ordinary primes: a story

Abstract

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 $k\in\{12$,$\,16$,$\,18$,$\,20$,$\,22$,$\,26\}$ 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 $p\leq 1\,000\,000$.