Algebraic results for the values $\vartheta_3(m\tau)$ and $\vartheta_3(n\tau)$ of the Jacobi theta-constant

Abstract

Let ${\vartheta }_3\left(\tau \right)=1+2{\sum }_{\nu =1}^{\infty }{e}^{\pi i{\nu }^2\tau }$ denote the classical Jacobi theta-constant. We prove that the two values ${\vartheta }_3\left(m\tau \right)$ and ${\vartheta }_3\left(n\tau \right)$ are algebraically independent over $\mathbb{Q}$ for any $\tau$ in the upper half-plane such that $q=e^{\pi i\tau }$ is an algebraic number, where $m,n\ge 2$ are distinct integers.