The tropical Abel--Prym map

TL;DR

This paper studies the properties of the tropical Abel--Prym map for hyperelliptic metric graphs, establishing its harmonicity, degree, and hyperelliptic nature, and explores the differences when the graphs are not hyperelliptic.

Contribution

It proves the harmonicity and degree of the tropical Abel--Prym map for hyperelliptic graphs and analyzes its injectivity and structure in non-hyperelliptic cases.

Findings

The tropical Abel--Prym map is harmonic of degree 2 for hyperelliptic graphs.

When the source graph is hyperelliptic, the Prym graph is also hyperelliptic with genus reduced by one.

The map often fails to be injective when the source graph is not hyperelliptic.

Abstract

We prove that the tropical Abel--Prym map $\Psi\colon \widetilde\Gamma\to Prym(\widetilde\Gamma/\Gamma)$ associated with a free double cover $\pi\colon \widetilde\Gamma\to \Gamma$ of hyperelliptic metric graphs is harmonic of degree $2$ in accordance with the already established algebraic result. We then prove a partial converse. Contrary to the analogous algebraic result, when the source graph of the double cover is not hyperelliptic, the Abel--Prym map is often not injective. When the source graph is hyperelliptic, we show that the Abel--Prym graph $\Psi(\widetilde\Gamma)$ is a hyperelliptic metric graph of genus $g_{\Gamma}-1$ whose Jacobian is isomorphic, as pptav, to the Prym variety of the cover. En route, we count the number of distinct free double covers by hyperelliptic metric graphs.