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.