Let denote the classical Jacobi theta-constant. We prove that the two values and are algebraically independent over for any in the upper half-plane such that is an algebraic number, where are distinct integers.
"Algebraic results for the values $\vartheta_3(m\tau)$ and $\vartheta_3(n\tau)$ of the Jacobi theta-constant." Mosc. J. Comb. Number Theory 8 (1) 71 - 79, 2019. https://doi.org/10.2140/moscow.2019.8.71