On Dihedral Group Actions on Riemann Surfaces

TL;DR

This paper explores dihedral group actions on compact Riemann surfaces, establishing a correspondence between geometric signatures and analytic representations, refining signature realization results, and applying findings to Jacobian decompositions.

Contribution

It provides a bijective correspondence between geometric signatures and analytic representations, refines existing signature realization results, and extends Jacobian decomposition classifications under dihedral actions.

Findings

Established a bijective correspondence between signatures and representations

Refined the signature realization theorem for dihedral actions

Classified Jacobians with dihedral symmetry and specific decompositions

Abstract

This article deals with dihedral group actions on compact Riemann surfaces and the interplay between different geometric data associated to them. First, a bijective correspondence between geometric signatures and analytic representations is obtained. Second, a refinement of a result of Bujalance, Cirre, Gamboa and Gromadzki about signature realization is provided. Finally, we apply our results to isogeny decompositions of Jacobians by Prym varieties and by elliptic curves, extending results of Carocca, Recillas and Rodr\'iguez. In particular, we give a complete classification of Jacobians with dihedral action whose group algebra decomposition induces a decomposition into factors of the same dimension.