On algebraic conditions for the non-vanishing of linear forms in Jacobi theta-constants

Elsner, Luca and Tachiya proved in [4] 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 [3] 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.