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A note on non-ordinary primes


Authors: Seokho Jin, Wenjun Ma and Ken Ono
Journal: Proc. Amer. Math. Soc. 144 (2016), 4591-4597
MSC (2010): Primary 11F33; Secondary 11F11
DOI: https://doi.org/10.1090/proc/13111
Published electronically: May 6, 2016
MathSciNet review: 3544511
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Abstract: Suppose that $ O_L$ is the ring of integers of a number field $ L$, and suppose that

$\displaystyle 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

$\displaystyle 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.

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Additional Information

Seokho Jin
Affiliation: School of Mathematics, Korea Institute for Advanced Study, Hoegiro 85, Dongdaemun-gu, Seoul 130-722, Korea
Email: seokhojin@kias.re.kr

Wenjun Ma
Affiliation: School of Mathematics, Shandong University, Jinan, Shandong, People’s Republic of China 250100
Email: wenjunma.sdu@hotmail.com

Ken Ono
Affiliation: Department of Mathematics, Emory University, Atlanta, Georgia 30322
Email: ono@mathcs.emory.edu

DOI: https://doi.org/10.1090/proc/13111
Received by editor(s): November 19, 2015
Received by editor(s) in revised form: January 7, 2016
Published electronically: May 6, 2016
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2016 American Mathematical Society