Level structures on tropical abelian varieties

We introduce level structures on tropical abelian varieties and give a modular interpretation of $A_g[m{]}^{\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}}$, the tropicalization of the moduli space ${\mathcal{A}}_g[m]$ of principally polarized abelian varieties with level $m$ structure. We study the case of abelian surfaces in greater depth. The link of $A_2[m{]}^{\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}}$ is an explicit simplicial complex whose vertices are the primitive vectors of $(\mathbb{Z}\mathrm{/}m\mathbb{Z}{)}^4$ up to sign. As a topological space, this link is homotopic to a wedge sum of closed orientable surfaces and circles; we compute the number of each of these and the genera of the surfaces. We deduce the weight zero compactly supported rational cohomology of ${\mathcal{A}}_2[m]$, completing a calculation of Oda-Schwermer from 1990.