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Proceedings of the American Mathematical Society

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

 

 

A transcendence measure for some special values of elliptic functions


Author: Robert Tubbs
Journal: Proc. Amer. Math. Soc. 88 (1983), 189-196
MSC: Primary 11J82; Secondary 11J89
DOI: https://doi.org/10.1090/S0002-9939-1983-0695238-2
MathSciNet review: 695238
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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\vert {P(\wp (\beta u))} \right\vert > - C{d^2}{t^2}{(\log t)^4}$.

(B) If $ \wp (z)$ does not have complex multiplication then $ \log \left\vert {P(\wp (\beta u))} \right\vert > - C{d^6}{t^2}{(\log t)^{14}}$.


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DOI: https://doi.org/10.1090/S0002-9939-1983-0695238-2
Keywords: Transcendental numbers, transcendence measure, elliptic functions
Article copyright: © Copyright 1983 American Mathematical Society