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 $θ_3(mτ)$ and $θ_3(nτ)$ are algebraically independent over $\mathbb{Q}$ for distinct integers $m,n$ under some conditions on $τ$. On the other hand, in 2018 Elsner and Tachiya also proved that three values $θ_3(mτ),θ_3(nτ)$ and $θ_3(\ell τ)$ are algebraically dependent over $\mathbb{Q}$. In this article we prove the non-vanishing of linear forms in $θ_3(mτ)$, $θ_3(nτ)$ and $θ_3(\ell τ)$ under various conditions on $m,n,\ell$, and $τ$. Among other things we prove that for odd and distinct positive integers $m,n>3$ the three numbers $θ_3(τ)$, $θ_3(mτ)$ and $θ_3(n τ)$ are linearly independent over $\overline{\mathbb{Q}}$ when $τ$ 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 $\mathbb{C(τ)}$ of the functions $θ_3(a_1 τ),\ldots,θ_3(a_m τ)$ for distinct positive rational numbers $a_1, \ldots a_m$ is also established.