Tropical covers, tropical abelian varieties and Prym varieties

We define and investigate the tropical Prym varieties associated to unramified Galois cyclic covers of tropical curves (or equivalently metric graphs) $\tilde{\mathrm{\Gamma }}\to \mathrm{\Gamma }$. 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 ${\mathbb{Z}}_3$-covers of tropical curves and their associated tropical Prym variety and compute its volume generalizing the case of double covers.