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

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发表日期：

2024年8月16日

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C. Elsner,
V. Kumar

摘要

Elsner, Luca and Tachiya proved in [4] that the values of the Jacobi-theta constants $\theta_3(m\tau)$ $θ_{3} (m τ)$ and $\theta_3(n\tau)$ $θ_{3} (n τ)$ are algebraically independent over $\mathbb{Q}$ $Q$ for distinct integers $m$ $m$ , $n$ $n$ under some conditions on $\tau$ $τ$ . On the other hand, in [3] Elsner and Tachiya also proved that three values $\theta_3(m\tau),\theta_3(n\tau)$ $θ_{3} (m τ), θ_{3} (n τ)$ and $\theta_3(\ell \tau)$ $θ_{3} (ℓ τ)$ are algebraically dependent over $\mathbb{Q}$ $Q$ . In this article we prove the non-vanishing of linear forms in $\theta_3(m\tau)$ $θ_{3} (m τ)$ , $\theta_3(n\tau)$ $θ_{3} (n τ)$ and $\theta_3(\ell \tau)$ $θ_{3} (ℓ τ)$ under various conditions on $m$ $m$ , $n$ $n$ , $\ell$ $ℓ$ , and $\tau$ $τ$ . Among other things we prove that for odd and distinct positive integers $m,n>3$ $m, n > 3$ the three numbers $\theta_3(\tau)$ $θ_{3} (τ)$ , $\theta_3(m\tau)$ $θ_{3} (m τ)$ and $\theta_3(n \tau)$ $θ_{3} (n τ)$ are linearly independent over $\overline{\mathbb{Q}}$ $\bar{Q}$ when $\tau$ $τ$ 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 $\mathbb{C(\tau)}$ $C (τ)$ of the functions $\theta_3(a_1 \tau), \dots, \theta_3(a_m \tau)$ $θ_{3} (a_{1} τ), \dots, θ_{3} (a_{m} τ)$ for distinct positive rational numbers $a_{1}, {\dots}, a_{m}$ $a_{1}, \dots, a_{m}$ is also established.