Wojciech J. Gajda

Abstract:

In joint work with Victoria Cantoral Farfán and John Voight we investigate families of even-dimensional Jacobians defined over Q and admitting an action of the quaternion group. Such abelian varieties are unusual in several ways: for example, their ring of algebraic cycles is not generated by divisor classes, a fact which has consequences both on their arithmetic and on their geometry. We prove that - for every even dimension greater than two - 100% of the members of the families we consider satisfy the Hodge, Tate and Mumford-Tate conjectures, compute their endomorphism rings, and provide explicit generators for their Hodge rings. As a consequence, we show that for such abelian varieties A 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.