Tropical covers, tropical abelian varieties and Prym varieties

TL;DR

This paper introduces tropical Prym varieties for unramified cyclic covers of tropical curves, extending classical concepts to the tropical setting and computing associated volumes for specific cases.

Contribution

It defines tropical Prym varieties for cyclic covers, studies their properties via group actions, and extends the Abel-Prym map to tropical cyclic covers.

Findings

Defined tropical Prym varieties for unramified Galois cyclic covers.

Extended the Abel-Prym map to tropical cyclic covers.

Computed the volume of tropical Prym varieties for free $\

Abstract

We define and investigate the tropical Prym varieties associated to unramified Galois cyclic covers of tropical curves (or equivalently metric graphs) $\tilde{\Gamma}\to \Gamma$. Our approach here is to study the tropical Prym varieties using group actions on tropical abelian varieties induced by the cyclic Galois group of the cover of tropical curves. We also define and conider the Abel-Prym map for tropical cyclic covers extending that for double covers. As a special case we consider free $\Z_3$-covers of tropical curves and their associated tropical Prym variety and compute its volume generalizing the case of double covers.