Families of Jacobians with quaternionic multiplication

In joint work with Victoria Cantoral-Farfán and John Voight we investigate explicit families of even-dimensional Jacobians defined over Q and admitting an action of the quaternion group. These abelian varieties are unusual in several ways: for example, the ring of their algebraic cycles is not generated by divisor classes, a fact which has consequences both on their arithmetic and on their geometry.

We prove that 100% of the members of the families we consider satisfy the Hodge, Tate and Mumford-Tate conjectures, and provide explicit generators for their Hodge rings. As a consequence, we show that, for every even dimension greater than 2, there exist infinitely many abelian varieties A such that the minimal field of definition of the endomorphisms and the minimal field over which the Galois representations attached to A have connected image are different.