Galois extensions and a Conjecture of Ogg

Abstract

Let $N=pq$ be a product of two distinct primes. There is an isogeny $J_0(N)^{\rm new}\to J^N$ defined over $\mathbf{Q}$ between the new quotient of $J_0(N)$ and the Jacobian of the Shimura curve attached to the indefinite quaternion algebra of discriminant $N$. In the case when $p=2,3,5,7,13$, Ogg made predictions about the kernels of these isogenies. We show that Ogg's conjecture is not true in general. Afterwards, we propose a strategy for proving results toward Ogg's conjecture in certain situations. Finally, we discuss this strategy in detail for $N=5\cdot 13$.