Approximation to the transcendental relationship of two algebraic points of the function $$\wp \left( z \right)$$ with complex multiplication

Abstract

For fixed ɛ>0, the following inequality holds: $$\left| {\frac{u}{\upsilon } - \wp } \right| > Cexp\left( { - \left( {lnH} \right)^{2 + \varepsilon } } \right)$$ for all numbers β belonging to a field K of finite degree over Q. The constant C>0 does not depend on β. H is the height of β. $$\wp $$ (u) and $$\wp $$ (v) are algebraic numbers, and u/v is a transcendental number. $$\wp $$ (z) is the Weierstrass function with complex multiplication and algebraic invariants. The proof is ineffective.