On transcendence of modulus of Jacobi elliptic functions

Анотація

Let sn₁ z, sn₂ z be the Jacobi elliptic functions, κ₁, κ₂ the moduli of these elliptic functions, 0<κ₁²<1, 0<κ₂²<1, τ₁,  τ₂, the values of modular variable, θ₃(τ₁), θ₃(τ₂) the theta constants. In this paper it is shown that there exists a transcendental number among κ₁, κ₂, θ₃(τ₁), θ₃(τ₂), if τ₁/τ₂ is irrational.