On the measure of algebraic independence of certain values of elliptic functions

Abstract

We provide a measure for the algebraic independence of some special values of the Weierstrass elliptic function with complex multiplication and algebraic invariants. Specifically, suppose ℘(z) is such an elliptic function, u is a nontorsion algebraic point for ℘(z), and β is an algebraic number which is cubic over the field of multiplications of ℘(z). We then give the following result: For every ε > 0 there exists a positive real number t( ε) > 0 such that for any nonzero integral polynomial P( X, Y), with t( P) = deg P + log height P > t( ε), log|P(℘(βu), ℘(β 2u))| > − exp(t(P) 4 + ε) .