Abelian varieties and Riemann surfaces with generalized quaternion group action

TL;DR

This paper studies Riemann surfaces and abelian varieties with automorphism groups isomorphic to generalized quaternion groups, providing decompositions, classifications, and explicit examples, extending known results in the field.

Contribution

It offers new isogeny decompositions, classifications of surfaces and Jacobians with quaternion group actions, and explicit constructions, extending previous work on quasiplatonic surfaces.

Findings

Decomposition of abelian varieties with quaternion group action

Classification of Riemann surfaces with such automorphisms

Explicit examples and period matrices for Jacobians

Abstract

In this article we consider Riemann surfaces and abelian varieties endowed with a group of automorphisms isomorphic to a generalized quaternion group. We provide isogeny decompositions of each abelian variety with this action, compute dimensions of the corresponding factors and provide conditions under which this decomposition is nontrivial. We then specialize our results to the case of Jacobians and relate them to the so-called genus-zero actions on Riemann surfaces. We also give a complete classification and description of the complex one-dimensional families of Riemann surfaces and Jacobians with a generalized quaternion group action, extending known results concerning the quasiplatonic case. Finally, we construct and describe 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.