# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## A transcendence measure for some special values of elliptic functionsHTML articles powered by AMS MathViewer

by Robert Tubbs
Proc. Amer. Math. Soc. 88 (1983), 189-196 Request permission

## 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 $\log \left | {P(\wp (\beta u))} \right | > - C{d^6}{t^2}{(\log t)^{14}}$.
References
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• Michel Waldschmidt, Transcendence methods, Queen’s Papers in Pure and Applied Mathematics, vol. 52, Queen’s University, Kingston, Ont., 1979. MR 633068
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