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

## TL;DR

This paper investigates specific isogenies between modular Jacobians and Shimura curves, confirming a conjecture by Ogg about the structure of the kernel, and providing explicit descriptions of these isogenies over ield.

## Contribution

It proves the existence of a particular isogeny with a kernel supported on Eisenstein ideals and describes the kernel explicitly, advancing understanding of Ribet's isogeny for $J_0(65)$.

## Key findings

- Existence of an isogeny with kernel supported on Eisenstein maximal ideals.
- The odd part of the kernel is generated by a cuspidal divisor of order 7.
- Confirmation of Ogg's conjecture regarding the kernel structure.

## 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 $J_0(65)\rightarrow J^{65}$ 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.