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Elliptic Functions and Transcendence

  • Michel WaldschmidtEmail author
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 17)

Transcendental numbers form a fascinating subject: so little is known about the nature of analytic constants that more research is needed in this area. Even when one is interested only in numbers like π and π that are related to the classical exponential function, it turns out that elliptic functions are required (so far, this should not last forever!) to prove transcendence results and get a better understanding of the situation. First we briefly recall some of the basic transcendence results related to the exponential function (Section l). Next, in Section 2, we survey the main properties of elliptic functions that are involved in transcendence theory. We survey transcendence theory of values of elliptic functions in Section 3, linear independence in Section 4, and algebraic independence in Section 5. This splitting is somewhat artificial but convenient. Moreover, we restrict ourselves to elliptic functions, even when many results are only special cases of statements valid for abelian functions. A number of related topics are not considered here (e.g., heights, p-adic theory, theta functions, Diophantine geometry on elliptic curves).

Transcendental numbers elliptic functions elliptic curves elliptic integrals algebraic independence transcendence measures measures of algebraic independence Diophantine approximation. 

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© Springer-Verlag New York 2008

Authors and Affiliations

  1. 1.Institut de Mathématiques de JussieuUniversité P. et M. Curie (Paris VI) UMR 7586 CNRS Problèmes Diophantiens Case 247ParisFrance

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