Algebraic independence results for values of Jacobi theta-constants

Abstract

Let $\theta_3(\tau)=1+2\sum_{\nu=1}^{\infty} q^{\nu^2}$ with $q=e^{i\pi \tau}$ and $\Im (\tau)>0$ denote the Thetanullwert of the Jacobi theta function \[\theta(z|\tau) \,=\,\sum_{\nu=-\infty}^{\infty} e^{\pi i\nu^2\tau + 2\pi i\nu z} \,.\] Moreover, let $\theta_2(\tau)=2\sum_{\nu=0}^{\infty} q^{{(\nu+1/2)}^2}$ and $\theta_4(\tau)=1+2\sum_{\nu=1}^{\infty} {(-1)}^{\nu}q^{\nu^2}$. 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 \[Q_n\Big( \,\frac{\theta_3^4(n\tau)}{\theta_3^4(\tau)},\frac{\theta_2^4(\tau)}{\theta_3^4(\tau)}\, \Big) \,=\, 0 \] holds for all complex numbers $\tau$ from the upper half plane of $\mathbb{C}$. These polynomials are used to prove the algebraic independence of $\theta_3(n\tau)$ and $\theta_3(\tau)$ for all algebraic numbers $q=e^{i\pi \tau}$ with $0<|q|<1$. Combining this with former results of the authors, it is shown that for such algebraic $q$ the numbers $\theta_3(n\tau)$ and $\theta_3(\tau)$ 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 $\theta_3(\ell\tau)$, $\theta_3(m\tau)$, and $\theta_3(n\tau)$ for integers $\ell,m,n\geq 1$ is also presented.