Algebraic independence results for values of Theta-constants, II

Let ${\theta }_3(\tau )=1+2{\sum }_{\nu =1}^{\mathrm{\infty }}{q}^{{\nu }^2}$ with $q=e^{i\pi \tau}$ denote the Thetanullwert of the Jacobi theta function $$\theta (z\mathrm{\mid }\tau )=\sum _{\nu =-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\pi i{\nu }^2\tau +2\pi i\nu z}\mathrm{.}$$ Moreover, let ${\theta }_2(\tau )=2{\sum }_{\nu =0}^{\mathrm{\infty }}{q}^{{(\nu +1\mathrm{/}2)}^2}$ and ${\theta }_4(\tau )=1+2{\sum }_{\nu =1}^{\mathrm{\infty }}{(-1)}^{\nu }{q}^{{\nu }^2}$. For algebraic numbers $q$ with $0<\mathrm{\mid }q\mathrm{\mid }<1$ and for any $j\in \{2,3,4\}$ we prove the algebraic independence over $\mathbb{Q}$ of the numbers ${\theta }_j(n\tau )$ and ${\theta }_j(\tau )$ for all odd integers $n\ge 3$. Assuming the same conditions on $q$ and $\tau$ as above, we obtain sufficient conditions by use of a criterion involving resultants in order to decide on the algebraic independence over $\mathbb{Q}$ of ${\theta }_j(2m\tau )$ and ${\theta }_j(\tau )$ $(j=2,3,4)$ and of ${\theta }_3(4m\tau )$ and ${\theta }_3(\tau )$ with odd positive integers $m$. In particular, we prove the algebraic independence of ${\theta }_3(n\tau )$ and ${\theta }_3(\tau )$ for even integers $n$ with $2\le n\le 22$. The paper continues the work of the first-mentioned author, who already proved the algebraic independence of ${\theta }_3(2^m\tau )$ and ${\theta }_3(\tau )$ for $m=1,2,\dots$.