On Algebraic Conditions for the Non-Vanishing of Linear Forms in Jacobi Theta-Constants

Abstract

Elsner, Luca and Tachiya proved in 2019 that the values of the Jacobi-theta constants $\theta_3(m\tau)$ and $\theta_3(n\tau)$ are algebraically independent over $\mathbb{Q}$ for distinct integers $m,n$ under some conditions on $\tau$. On the other hand, in 2018 Elsner and Tachiya also proved that three values $\theta_3(m\tau),\theta_3(n\tau)$ and $\theta_3(\ell \tau)$ are algebraically dependent over $\mathbb{Q}$. In this article we prove the non-vanishing of linear forms in $\theta_3(m\tau)$, $\theta_3(n\tau)$ and $\theta_3(\ell \tau)$ under various conditions on $m,n,\ell$, and $\tau$. Among other things we prove that for odd and distinct positive integers $m,n>3$ the three numbers $\theta_3(\tau)$, $\theta_3(m\tau)$ and $\theta_3(n \tau)$ are linearly independent over $\overline{\mathbb{Q}}$ when $\tau$ is an algebraic number of some degree greater or equal to 3. In some sense this fills the gap between the above-mentioned former results on theta constants. A theorem on the linear independence over $\mathbb{C(\tau)}$ of the functions $\theta_3(a_1 \tau),\ldots,\theta_3(a_m \tau)$ for distinct positive rational numbers $a_1, \ldots a_m$ is also established.