In this talk, I will discuss the structure of tropical and non-Archimedean analytic genus two curves and their moduli. Abstractly, they correspond to weighted genus two metric graphs. In the classical realm, such curves can be viewed as double covers of the projective line, branched at six points. The valuations of these branch points, provide combinatorial rules to characterize the corresponding metric graphs together with a harmonic map to a metric tree on six leaves. A second description is given by the planar hyperelliptic equation. Even though this tropicalization shows no genus, we will show explicit re-embeddings of the input planar embedding that reveals the correct metric graphs, extending previous joint work with Hannah Markwig on elliptic curves.

Finally, we consider the moduli space of abstract genus two tropical curves and translate the classical Igusa invariants characterizing isomorphism classes of genus two algebraic curves into the tropical realm. While these tropical Igusa functions do not yield coordinates on the tropical moduli space, we propose an alternative set of invariants that provides new length data. This is joint work with Hannah Markwig (arXiv:1801.00378).