The tropical Poincaré–Prym formula

Let $\widetilde{\Gamma} \to \Gamma$ be a possibly dilated double cover of tropical curves, and fix a base point $q\in \widetilde{\Gamma}$. Set $g_0=\dim \operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$, let $1\leq d\leq g_0$, and let $\widetilde{Y}_d$ be the image of the $d$-fold Abel–Prym map

$$\psi_q^d:\widetilde{\Gamma}^d\to \operatorname{Prym}(\widetilde{\Gamma}/\Gamma).$$

The tropical Poincaré–Prym formula. The cycle class of $\widetilde{Y}_d$ satisfies

$$[\widetilde{Y}_d]=\frac{2^d}{(g_0-d)!}[\Xi]^{g_0-d}\in H_{d,d}\bigl(\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)\bigr),$$

where $[\Xi]$ is the class of the principal polarization of $\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$. This is the tropical analogue of the Poincaré–Prym formula for Prym varieties; the paper proves the formula for $d=1$, while the assertion for all $1\leq d\leq g_0$ remains open.