Some differential equations for the Riemann $\theta$-function on Jacobians

Abstract

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. Moreover, we give a generalization of Rosenhain's formula to hyperelliptic Riemann surfaces as conjectured by Gu\`ardia.Comment: Comments are welcom