On Ribet’s isogeny for 𝐽₀(65)

Klosin, Krzysztof; Papikian, Mihran

doi:10.1090/proc/14019

On Ribet’s isogeny for $J_0(65)$
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by Krzysztof Klosin and Mihran Papikian PDF

Proc. Amer. Math. Soc. 146 (2018), 3307-3320 Request permission

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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