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 \[θ(z|τ) \,=\,\sum_{ν=-\infty}^{\infty} e^{πiν^2τ+ 2πiνz} \,.\] 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 $n\ge 6$, 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 $\tau$ 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 $n\ge 2$. 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 $\ell ,m,n\ge 1$ is also presented.

arXiv comment19 pages

References in corpus (1)

• Algebraic independence results for values of Theta-constants, II