On the transcendence of moduli of the Jacobian elliptic functions

Let $\mathrm{sn}_1z$ and $\mathrm{sn}_2z$ be the Jacobian elliptic functions of moduli $\varkappa_1$ and $\varkappa_2$, $0<\varkappa_1^2<1$, $0<\varkappa_2^2<1$, $\tau_1$ 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.