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 θ 3 (m τ) and θ 3 (n τ) are algebraically independent over Q for distinct integers m , n under some conditions on τ . On the other hand, in [3] Elsner and Tachiya also proved that three values θ 3 (m τ) , θ 3 (n τ) and θ 3 (ℓ τ) are algebraically dependent over Q . In this article we prove the non-vanishing of linear forms in θ 3 (m τ) , θ 3 (n τ) and θ 3 (ℓ τ) under various conditions on m , n , ℓ , 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 (m τ) and θ 3 (n τ) are linearly independent over 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 C (τ) of the functions θ 3 (a 1 τ) , ⋯ , θ 3 (a m τ) for distinct positive rational numbers a 1 , ⋯ , a m is also established. [ABSTRACT FROM AUTHOR]