TY - JOUR
AU - Carsten Elsner
AU - Yohei Tachiya
PY - 2018/09/05
Y2 - 2019/02/23
TI - Algebraic results for certain values of the Jacobi theta-constant $\theta_3(\tau)$
JF - MATHEMATICA SCANDINAVICA
JA - MathScand
VL - 123
IS - 2
SE - Articles
DO - 10.7146/math.scand.a-105465
UR - https://www.mscand.dk/article/view/105465
AB - In its most elaborate form, the Jacobi theta function is defined for two complex variables $z$ and τ by $\theta (z|\tau ) =\sum _{
u =-\infty }^{\infty } e^{\pi i
u ^2\tau + 2\pi i
u z}$, which converges for all complex number $z$, and τ in the upper half-plane. The special case \[ \theta _3(\tau ):=\theta (0|\tau )= 1+2\sum _{
u =1}^{\infty } e^{\pi i
u ^2 \tau } \] is called a Jacobi theta-constant or Thetanullwert of the Jacobi theta function $\theta (z|\tau )$. In this paper, we prove the algebraic independence results for the values of the Jacobi theta-constant $\theta _3(\tau )$. For example, the three values $\theta _3(\tau )$, $\theta _3(n\tau )$, and $D\theta _3(\tau )$ are algebraically independent over $\mathbb{Q} $ for any τ such that $q=e^{\pi i\tau }$ is an algebraic number, where $n\geq 2$ is an integer and $D:=(\pi i)^{-1}{d}/{d\tau }$ is a differential operator. This generalizes a result of the first author, who proved the algebraic independence of the two values $\theta _3(\tau )$ and $\theta _3(2^m\tau )$ for $m\geq 1$. As an application of our main theorem, the algebraic dependence over $\mathbb{Q} $ of the three values $\theta _3(\ell \tau )$, $\theta _3(m\tau )$, and $\theta _3(n\tau )$ for integers $\ell ,m,n\geq 1$ is also presented.
ER -