Moscow Mathematical Journal

Volume 26, Issue 1, January–March 2026 pp. 19–50.

On Dihedral Group Actions on Riemann Surfaces

Authors: Pablo Alvarado-Seguel (1) and Sebastián Reyes-Carocca (2)
Author institution:(1) Instituto de Ciencias Matemáticas, C/ Nicolás Cabrera 13-15, 28049, Madrid, España
(2) Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Las Palmeras 3425, Santiago, Chile

Summary:

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 Rodríguez. 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.

2020 Math. Subj. Class. 30F10, 14H37, 14H40, 14H30.

Keywords: Riemann surfaces, Jacobians and Prym varieties, group actions.

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