Abstract:

Let be a product of two distinct primes. There is an isogeny defined over between the new quotient of and the Jacobian of the Shimura curve attached to the indefinite quaternion algebra of discriminant . In the case when , Andrew 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 . (This is a joint work with K. Klosin.)