A note on non-ordinary primes
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- by Seokho Jin, Wenjun Ma and Ken Ono
- Proc. Amer. Math. Soc. 144 (2016), 4591-4597
- DOI: https://doi.org/10.1090/proc/13111
- Published electronically: May 6, 2016
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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.References
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Bibliographic Information
- Seokho Jin
- Affiliation: School of Mathematics, Korea Institute for Advanced Study, Hoegiro 85, Dongdaemun-gu, Seoul 130-722, Korea
- MR Author ID: 1029608
- Email: seokhojin@kias.re.kr
- Wenjun Ma
- Affiliation: School of Mathematics, Shandong University, Jinan, Shandong, People’s Republic of China 250100
- MR Author ID: 1125977
- Email: wenjunma.sdu@hotmail.com
- Ken Ono
- Affiliation: Department of Mathematics, Emory University, Atlanta, Georgia 30322
- MR Author ID: 342109
- Email: ono@mathcs.emory.edu
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4591-4597
- MSC (2010): Primary 11F33; Secondary 11F11
- DOI: https://doi.org/10.1090/proc/13111
- MathSciNet review: 3544511