A Note On Non-ordinary Primes

TL;DR

This paper proves the existence of normalized Hecke eigenforms that are non-ordinary at any finite set of primes, using an elementary approach based on a generalization of previous work.

Contribution

It establishes the existence of eigenforms non-ordinary at multiple primes simultaneously, extending prior results through a simplified proof.

Findings

Existence of eigenforms non-ordinary at any finite prime set

Elementary proof based on a generalization of earlier work

Applicable to all primes in a finite set

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: $q := e^{2\pi iz}$) 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 $\mathfrak{p}\subset O_L$ above $p$ for which $a_f(p)\equiv 0 \ (mod\ {\mathfrak{p}})$. 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.