Galois extensions and a Conjecture of Ogg

Krzysztof Klosin, Mihran Papikian

Published 2018-12-14Version 1

Let $N=pq$, where $p=2,3,5,7,13$ and $q\neq p$ is another prime. We propose a general strategy for proving a conjecture of Ogg about isogenies from the new quotient of $J_0(N)$ to the Jacobian of the Shimura curve attached to the indefinite quaternion algebra over $\mathbf{Q}$ of discriminant $N$. This strategy is based on the results of Helm and Ribet. Using this strategy, we prove a general conditional result toward Ogg's conjecture and discuss in detail the case when $N=65$.