Algebraic independence results for values of Jacobi theta-constants

Let $θ_3(τ)=1+2\sum_{ν=1}^{\infty} q^{ν^2}$ with $q=e^{iπτ}$ and $\Im (τ)>0$ 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 every even integer $n\geq 6$, which is not a power of two, we prove constructively the existence of a nontrivial integer polynomial $Q_n(X,Y)$ such that $Q_n\Big( \,\frac{θ_3^4(nτ)}{θ_3^4(τ)},\frac{θ_2^4(τ)}{θ_3^4(τ)}\, \Big) \,=\, 0$ holds for all complex numbers $τ$ from the upper half plane of $\mathbb{C}$. These polynomials are used to prove the algebraic independence of $θ_3(nτ)$ and $θ_3(τ)$ for all algebraic numbers $q=e^{iπτ}$ with $0<|q|<1$. Combining this with former results of the authors, it is shown that for such algebraic $q$ the numbers $θ_3(nτ)$ and $θ_3(τ)$ are algebraically independent over $\mathbb{Q}$ for every integer $n\geq 2$. A result on the algebraic dependence over $\mathbb{Q}$ of the three numbers $θ_3(\ellτ)$, $θ_3(mτ)$, and $θ_3(nτ)$ for integers $\ell,m,n\geq 1$ is also presented.

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