Abelian varieties and Riemann surfaces with generalized quaternion group action

Abstract

We consider abelian varieties with a group of automorphisms isomorphic to a generalized quaternion group. We provide isogeny decompositions, compute dimensions of the corresponding factors and give conditions under which this decomposition is nontrivial. We then specialize our results to Jacobians and relate them to the so-called genus-zero actions on Riemann surfaces. A complete classification of uniparametric families of Riemann surfaces with a generalized quaternion group action is given, extending known results for the quasiplatonic case. Finally, we construct explicit families of abelian varieties with a quaternion group action and derive a period matrix for the Jacobian of the surface with full automorphism group of second largest order among the hyperelliptic surfaces of genus four.