GALOIS EXTENSIONS and A CONJECTURE of OGG

Let N = pq be a product of two distinct primes. There is an isogeny J0(N)new → JN defined over Q between the new quotient J0(N) and the Jacobian of the Shimura curve attached to 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 false 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• 13.