Differential geometry of the Fermat quartic and theta functions

The universal curve over a finite cover of the moduli space of elliptic curves with level four structure is embedded in CP³ as the Fermat quartic and is parametrized via the four Jacobi theta functions. Constructions from completely integrable systems have shown the importance of looking at the curvature of certain spaces and here we compute sectional curvatures. For our computations, we choose the ambient Fubini-Study metric of CP³. We also derive several theta identities which arise from the quartic’s holomorphic two-form.