Simultaneous approximation of values of Weierstrass and Jacobi elliptic functions in the periods and algebraic point

Анотація

Let $\wp \left(Z\right)$, $sn\left(z\right)$ be algebraically independent Weierstrass and Jacobi elliptic functions with algebraic invariants and algebraic elliptic module, $\left(2{\omega }_1,2{\omega }_3\right)$ and $\left(4K, 2iK\text{'}\right)$ be the main periods of $\wp \left(Z\right)$ and $sn\left(z\right)$ respectively, $\alpha$ be an  algebraic number different from the poles of of $\wp \left(Z\right)$ and $sn\left(z\right)$. We estimate from below the simultaneous approximation of $sn\left(2{\omega }_1\right)$, $sn\left(\alpha \right)$, $\wp \left(4k\right)$, and $\wp \left(\alpha \right)$.