The tropical Abel--Prym map

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.