Algebraic independence of values of elliptic functions (Q1065854)

Several authors have given results showing that among numbers of the form \(e^{uv}\) there are at least two algebraically independent. In this paper we give a complete set of analogous results for numbers \(\wp (uv)\), where \(\wp (z)\) is a Weierstraß elliptic function with algebraic invariants. For example, let \(u_ 1,...,u_ n\) be complex numbers linearly independent over the field K of complex multiplications of \(\wp (z)\), and let \(v_ 1,...,v_ m\) be complex numbers linearly independent over \({\mathbb{Q}}\). Then among the \(u_ i\), \(\wp (u_ iv_ j)\) (1\(\leq i\leq n\), \(1\leq j\leq m)\) there are at least two algebraically independent numbers provided \(mn\geq 2m+2n\). This implies the algebraic independence of \(\wp (\beta u)\), \(\wp (\beta^ 2u)\) whenever \(\wp (u)\) is algebraic and \(\beta\) is cubic over K; which is the elliptic analogue of a well- known result of Gelfond (1949). The proofs involve refinements of the zero estimates [Invent. Math. 64, 489-516 (1981; Zbl 0467.10025)] of the authors, to appear soon in the same journal.