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 ${\theta }_3(a_1\tau ),\dots ,{\theta }_3(a_m\tau )$ for distinct positive rational numbers $a_1,\dots a_m$ is also established.