Diophantine approximations for a constant related to elliptic functions

Abstract

This paper is devoted to the study of rational approximations of the ratio $\eta \left(\lambda \right)/\omega \left(\lambda \right)$, where $\omega \left(\lambda \right)$ and $\eta \left(\lambda \right)$ are the real period and real quasi-period, respectively, of the elliptic curve $y^2=x(x-1)(x-\lambda )$. Using monodromy principle for hypergeometric function in the logarithm case we obtain rational approximations of $(\eta /\omega )\left(\lambda \right)$ with $\lambda ∊Q$ and we shall find new measures of irrationality, both in the archimedean and non archimedean case.