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On Ribet's isogeny for $ J_0(65)$

Authors: Krzysztof Klosin and Mihran Papikian
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 11G18
Published electronically: February 28, 2018
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Abstract: Let $ J^{65}$ be the Jacobian of the Shimura curve attached to the indefinite quaternion algebra over $ \mathbb{Q}$ of discriminant $ 65$. We study the isogenies $ J_0(65)\to J^{65}$ defined over $ \mathbb{Q}$, whose existence was proved by Ribet. We prove that there is an isogeny whose kernel is supported on the Eisenstein maximal ideals of the Hecke algebra acting on $ J_0(65)$, and, moreover, the odd part of the kernel is generated by a cuspidal divisor of order $ 7$, as is predicted by a conjecture of Ogg.

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Additional Information

Krzysztof Klosin
Affiliation: Department of Mathematics, Queens College, City University of New York, 65-30 Kissena Boulevard Flushing, New York 11367

Mihran Papikian
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802

Keywords: Modular curves, Ribet's isogeny, Eisenstein ideal, cuspidal divisor group
Received by editor(s): July 19, 2017
Received by editor(s) in revised form: November 13, 2017
Published electronically: February 28, 2018
Additional Notes: The first author was supported by the Young Investigator Grant #H98230-16-1-0129 from the National Security Agency, and by a PSC-CUNY award jointly funded by the Professional Staff Congress and the City University of New York.
The second author was partially supported by grants from the Simons Foundation (245676) and the National Security Agency (H98230-15-1-0008).
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2018 American Mathematical Society

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