The tropical Abel–Prym map

We prove that, under mild assumptions, the tropical Abel–Prym map Ψ : Γ ˜ → Prym ( Γ ˜ / Γ ) associated with a free double cover π : Γ ˜ → Γ is harmonic of degree 2 if and only if the source graph Γ ˜ is hyperelliptic. This is in accordance with the already established algebraic result. In this case, the Abel–Prym graph Ψ ( Γ ˜ ) is hyperelliptic of genus g Γ - 1 and its Jacobian is isomorphic, as a pptav, to the Prym variety of the cover. We further show that the Abel–Prym graph coincides with a connected component of the tropical bigonal construction. En route, we count the number of distinct free double covers by hyperelliptic metric graphs.