Some Differential Equations for the Riemann θ-Function on Jacobians

This is the authors' abstract. We don't add key points for this paper.

We prove some differential equations for the Riemann theta function associated to the Jacobian of a Riemann surface. The proof is based on some variants of a formula by Fay for the theta function, which are motivated by their analogues in Arakelov theory of Riemann surfaces.