Geodesic
On the transcendence of moduli of the Jacobian elliptic functions
Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 6, pp. 167-172 
Ya. M. Kholyavka. On the transcendence of moduli of the Jacobian elliptic functions. Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 6, pp. 167-172. http://geodesic.mathdoc.fr/item/FPM_2010_16_6_a12/
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     title = {On the transcendence of moduli of the {Jacobian} elliptic functions},
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\mathrm{sn}_1z$ and $\mathrm{sn}_2z$ be the Jacobian elliptic functions of moduli $\varkappa_1$ and $\varkappa_2$, $0\varkappa_1^21$, $0\varkappa_2^21$, $\tau_1$ and $\tau_2$ be the values of the modular variable, $\theta_3(\tau_1)$ and $\theta_3(\tau_2)$ be the theta constants. In this paper, the set $\varkappa_1$, $\varkappa_2$, $\theta_3(\tau_1)$, and $\theta_3(\tau_2)$ is shown to contain a transcendental number, provided that $\tau_1/\tau_2$ is irrational.

[1] Nesterenko Yu. V., “Ob arifmeticheskikh svoistvakh znachenii teta-konstant”, Fundament. i prikl. mat., 11:6 (2005), 95–122 | MR | Zbl

[2] Uitteker E. E., Vatson Dzh. N., Kurs sovremennogo analiza, v. 2, Fizmatgiz, M., 1963

[3] Kholyavka Ya. M., “O mere algebraicheskoi nezavisimosti znachenii ellipticheskikh funktsii Yakobi”, Fundament. i prikl. mat., 11:6 (2005), 209–219 | MR | Zbl

[4] Shidlovskii A. B., Transtsendentnye chisla, Nauka, M., 1987 | MR

[5] Fel'dman N. I., Nesterenko Yu. V., Transcendental Numbers, Springer, Berlin, 1998 | MR | Zbl

[6] Masser D., Elliptic Functions and Transcendence, Lect. Notes Math., 437, Springer, Berlin, 1975 | MR | Zbl

[7] Waldschmidt M., Elliptic functions and transcendence http://www.math.jussieu.fr/~miw/