On Jacobians with group action and coverings

TL;DR

This paper investigates how the group algebra decomposition of Jacobian varieties, associated with group actions on Riemann surfaces, can be lifted through regular coverings under certain conditions.

Contribution

It establishes conditions under which the group algebra decomposition of Jacobians can be lifted via regular coverings, extending known decomposition results.

Findings

Decomposition can be lifted under specific conditions

Provides criteria for lifting Jacobian decompositions

Extends the understanding of Jacobian structures in coverings

Abstract

Let $S$ be a compact Riemann surface and let $H$ be a finite group. It is known that if $H$ acts on $S$ then there is a $H$-equivariant isogeny decomposition of the Jacobian variety $JS$ of $S,$ called the group algebra decomposition of $JS$ with respect to $H.$ If $S_1 \to S_2$ is a regular covering map, then it is also known that the group algebra decomposition of $JS_1$ induces an isogeny decomposition of $JS_2.$ In this article we deal with the converse situation. More precisely, we prove that the group algebra decomposition can be lifted under regular covering maps, under appropriate conditions.