## A generalization of Jacobi's derivative formula to dimension two, II

### Volume 190 / 2019

#### Abstract

We give a new level-1 generalization of Rosenhain’s derivative formula for theta functions in two variables. Namely, we show that for $\tau$ in the degree-2 Siegel upper half-space, the jacobian at $0$ of an odd theta function $\theta[\delta](z,\tau)$ in two variables $z=\big({{z_1}\atop {z_2}}\big)$ with the numerator of its logarithmic Hessian, $X[\delta](z,\tau)$, gives a constant times the genus-2 level-1 Siegel modular form (with character) of weight 5. The gradient of $\theta[\delta](z,\tau)$ is a vector-valued modular form and we modify $X[\delta](z,\tau)$ by the addition of a multiple of $\theta[\delta](z,\tau)$ times a Siegel quasimodular form, so that its gradient at 0 is a vector-valued modular form as well. These formulas complement the results in the precursor paper from 1988 and will play a crucial role in an upcoming article on “modular models” for jacobians of curves of genus 2, and we discuss their geometric and arithmetic significance.