A transcendence measure for some special values of elliptic functions

Abstract:

Among T. Schneider’s results is the following: Let $\wp (z)$ be the Weierstrass elliptic function with algebraic invariants. If $\wp (u)$ and $\beta$ are both algebraic, $\beta \notin {K_\tau }$, then $\wp (\beta u)$ is transcendental. In this paper we provide a transcendence measure for this value. Let $P(X)$ be a nonzero polynomial, with integral coefficients, of degree $d$ and height $h$, and put $t = d + \log h$. Then there is an effectively computable constant $C$, which does not depend on $P(X)$. such that: (A) If $\wp (z)$ has complex multiplication then $\log \left | {P(\wp (\beta u))} \right | > - C{d^2}{t^2}{(\log t)^4}$. (B) If $\wp (z)$ does not have complex multiplication then $\log \left | {P(\wp (\beta u))} \right | > - C{d^6}{t^2}{(\log t)^{14}}$.