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

TL;DR

This paper investigates algebraic independence and linear independence of Jacobi theta constants at various arguments, establishing new conditions under which linear forms do not vanish, thus filling gaps in the understanding of their algebraic relations.

Contribution

It proves the non-vanishing of linear forms in theta constants for specific parameters, extending previous results and establishing linear independence over algebraic closures and function fields.

Findings

Proves linear independence of ( au), (m), (n) for odd, distinct m,n>3 when  is algebraic of degree  or more.

Establishes linear independence over () of theta functions at rational multiples of .

Fills gaps between previous algebraic independence and dependence results for 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.