Decomposition of Jacobian varieties of curves with dihedral actions via equisymmetric stratification

TL;DR

This paper investigates how the Jacobian varieties of certain Riemann surfaces with dihedral group actions decompose, providing explicit models and analyzing their structure through topological and algebraic methods.

Contribution

It offers a detailed analysis of the group algebra decomposition of Jacobians for surfaces with dihedral symmetries, including explicit models and the computation of Shimura domain dimensions.

Findings

Decomposition of Jacobians into factors related to intermediate coverings

Explicit affine models for Jacobian decompositions

Calculation of Shimura domain dimensions for the families studied

Abstract

Given a compact Riemann surface $X$ with an action of a finite group $G$, the group algebra Q[G] provides an isogenous decomposition of its Jacobian variety $JX$, known as the group algebra decomposition of $JX$. We consider the set of equisymmetric Riemann surfaces $\mathcal{M}(2n-1, D_{2n}, \theta)$ for all $n\geq 2$. We study the group algebra decomposition of the Jacobian $JX$ of every curve $X\in \mathcal{M}(2n-1, D_{2n},\theta)$ for all admissible actions, and we provide affine models for them. We use the topological equivalence of actions on the curves to obtain facts regarding its Jacobians. We describe some of the factors of $JX$ as Jacobian (or Prym) varieties of intermediate coverings. Finally, we compute the dimension of the corresponding Shimura domains.