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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On rational Eisenstein primes and the rational cuspidal groups of modular Jacobian varieties


Author: Hwajong Yoo
Journal: Trans. Amer. Math. Soc. 372 (2019), 2429-2466
MSC (2010): Primary 11F33, 11F80, 11G18
DOI: https://doi.org/10.1090/tran/7645
Published electronically: April 18, 2019
MathSciNet review: 3988582
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Abstract: Let $N$ be a non-squarefree positive integer and let $\ell$ be an odd prime such that $\ell ^2$ does not divide $N$. Consider the Hecke ring $\mathbb {T}(N)$ of weight $2$ for $\Gamma _0(N)$ and its rational Eisenstein primes of $\mathbb {T}(N)$ containing $\ell$. If $\mathfrak {m}$ is such a rational Eisenstein prime, then we prove that $\mathfrak {m}$ is of the form $(\ell , ~\mathcal {I}^D_{M, N})$, where we also define the ideal $\mathcal {I}^D_{M, N}$ of $\mathbb {T}(N)$. Furthermore, we prove that $\mathcal {C}(N)[\mathfrak {m}] \neq 0$, where $\mathcal {C}(N)$ is the rational cuspidal group of $J_0(N)$. To do this, we compute the precise order of the cuspidal divisor $\mathcal {C}^D_{M, N}$ and the index of $\mathcal {I}^D_{M, N}$ in $\mathbb {T}(N)\otimes \mathbb {Z}_\ell$.


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Additional Information

Hwajong Yoo
Affiliation: Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang, Republic of Korea 37673
Address at time of publication: College of Liberal Studies, Seoul National University, Gwanak-Ro 1, Gwanak-Gu, Seoul, 08826, South Korea
MR Author ID: 1146780
Email: hwajong@snu.ac.kr

Keywords: Cuspidal group, Eisenstein ideals
Received by editor(s): September 11, 2017
Received by editor(s) in revised form: May 3, 2018, and May 17, 2018
Published electronically: April 18, 2019
Additional Notes: This work was supported by IBS-R003-D1
Article copyright: © Copyright 2019 American Mathematical Society