Rational torsion points on Jacobians of Shimura curves

Abstract

Let p and q be distinct primes. Consider the Shimura curve Xpq associated to the indefinite quaternion algebra of discriminant pq over Q. Let Jpq be the Jacobian variety of Xpq, which is an abelian variety over Q. For an odd prime l, we provide sufficient conditions for the non-existence of rational points of order l on Jpq. As an application, we find some non-trivial subgroups of the kernel of an isogeny from the new quotient J0(pq) new of J0(pq) to Jpq. © 2016 London Mathematical Society