Algebraic results for certain values of the Jacobi theta-constant $\theta_3(\tau)$


  • Carsten Elsner
  • Yohei Tachiya



In its most elaborate form, the Jacobi theta function is defined for two complex variables $z$ and τ by $\theta (z|\tau ) =\sum _{\nu =-\infty }^{\infty } e^{\pi i\nu ^2\tau + 2\pi i\nu z}$, which converges for all complex number $z$, and τ in the upper half-plane. The special case \[ \theta _3(\tau ):=\theta (0|\tau )= 1+2\sum _{\nu =1}^{\infty } e^{\pi i\nu ^2 \tau } \] 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.


Bertrand, D., Theta functions and transcendence, Ramanujan J. 1 (1997), no. 4 (International Symposium on Number Theory, Madras, 1996), 339–350.

Duverney, D., Nishioka, K., Nishioka, K., and Shiokawa, I., Transcendence of Jacobi's theta series, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), no. 9, 202–203.

Elsner, C., Algebraic independence results for values of theta-constants, Funct. Approx. Comment. Math. 52 (2015), no. 1, 7–27.

Elsner, C., Shimomura, S., and Shiokawa, I., Algebraic independence results for reciprocal sums of Fibonacci numbers, Acta Arith. 148 (2011), no. 3, 205–223.

Lang, S., Elliptic functions, second ed., Graduate Texts in Mathematics, vol. 112, Springer-Verlag, New York, 1987.

Lawden, D. F., Elliptic functions and applications, Applied Mathematical Sciences, vol. 80, Springer-Verlag, New York, 1989.

Nesterenko, Y. V., Modular functions and transcendence questions, Mat. Sb. 187 (1996), no. 9, 65–96.

Nesterenko, Y. V., On some identities for theta-constants, in “Diophantine analysis and related fields 2006”, Sem. Math. Sci., vol. 35, Keio Univ., Yokohama, 2006, pp. 151--160.

Nesterenko, Y. V., Algebraic independence, Tata Institute of Fundamental Research, Bombay, Narosa Publishing House, New Delhi, 2009.




How to Cite

Elsner, C., & Tachiya, Y. (2018). Algebraic results for certain values of the Jacobi theta-constant $\theta_3(\tau)$. MATHEMATICA SCANDINAVICA, 123(2), 249–272.