Algebraic independence results for values of Theta-constants, II

Let $θ_3(τ)=1+2\sum_{ν=1}^{\infty} q^{ν^2}$ with $q=e^{iπτ}$ denote the Thetanullwert of the Jacobi theta function $θ(z|τ) \,=\,\sum_{ν=-\infty}^{\infty} e^{πiν^2τ+ 2πiνz} \,.$ Moreover, let $θ_2(τ)=2\sum_{ν=0}^{\infty} q^{{(ν+1/2)}^2}$ and $θ_4(τ)=1+2\sum_{ν=1}^{\infty} {(-1)}^νq^{ν^2}$. For algebraic numbers $q$ with $0<|q|<1$ and for any $j\in \{ 2,3,4\}$ we prove the algebraic independence over $\mathbb{Q}$ of the numbers $θ_j(nτ)$ and $θ_j(τ)$ for all odd integers $n\geq 3$. Assuming the same conditions on $q$ and $τ$ 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 $θ_j(2mτ)$ and $θ_j(τ)$ $(j=2,3,4)$ and of $θ_3(4mτ)$ and $θ_3(τ)$ with odd positive integers $m$. In particular, we prove the algebraic independence of $θ_3(nτ)$ and $θ_3(τ)$ for even integers $n$ with $2\leq n\leq 22$. The paper continues the work of the first-mentioned author, who already proved the algebraic independence of $θ_3(2^mτ)$ and $θ_3(τ)$ for $m=1,2,\dots$.

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