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Approximation et indépendance algébrique de quasi-périodes de variétés abéliennes

Abstract : Periods and ``quasi-periods'' (a.k.a., resp., periods of the first and second kind) of an abelian variety $A$ defined over a subfield of $\CC$ are obtained by integrating, along closed paths on $A(\CC)$, rational differentials on $A$, meromorphic and without residues so that these integrals are well-defined; the first kind is obtained by considering only regular (holomorphic) differentials. Our focus, in the first part of the thesis, is on the ``modular method'' developed by Barré, Diaz, Gramain, Philibert and Nesterenko; we use and somewhat refine it to obtain, in particular, an algebraic approximation measure for the ratio of a period of an elliptic curve defined over $\bar\QQ$ by its associated quasi-period; this improves on a recent result of N.\,Saradha, allowing it to almost include that obtained in 1980 by Reyssat using the ``elliptic method''. Part two revolves around various possible extensions of Chudnovsky's theorems (from the 70's) on algebraic independence of quasi-periods of elliptic curves; these include extensions to abelian varieties of any dimension, as well as results on simultaneous (algebraic) approximation refining assertions about algebraic independence. Somehow, although quite different in nature, these two topics meet up at one point as they can both benefit from a trick suggested by Chudnovsky in the early '80s, namely to exhibit and take advantage of the ``G-function property'' (or ``Eisenstein condition'' in Pólya and Szegö) in arithmetic estimates for the transcendence proof; to this end we use, in the first part, generalizations to several variables of Eisenstein's theorem and Weierstrass's sigma function used by Chudnovsky, and in the second part, links between modular (especially theta) and hypergeometric functions.
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Submitted on : Monday, April 29, 2002 - 1:56:30 AM
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Pierre Grinspan. Approximation et indépendance algébrique de quasi-périodes de variétés abéliennes. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2000. Français. ⟨tel-00001328⟩

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