A note on non-ordinary primes

Abstract:

Suppose that $O_L$ is the ring of integers of a number field $L$, and suppose that \[ f(z)=\sum _{n=1}^{\infty }a_f(n)q^n\in S_k\cap O_L[[q]] \] (note: $q:=e^{2\pi i z}$) is a normalized Hecke eigenform for $\mathrm {SL}_2(\mathbb {Z})$. We say that $f$ is non-ordinary at a prime $p$ if there is a prime ideal $\frak {p}\subset O_L$ above $p$ for which \[ a_f(p)\equiv 0\pmod {\frak {p}}. \] For any finite set of primes $S$, we prove that there are normalized Hecke eigenforms which are non-ordinary for each $p\in S$. The proof is elementary and follows from a generalization of work of Choie, Kohnen and the third author.