# Contributions à l'étude diophantienne des polylogarithmes et des groupes algébriques

Abstract : The first part of this thesis is dedicated to irrationality of values of polylogarithms. First, we exhibit changes of variables between multiple integrals, which allow us to generalize Rhin-Viola's groups and to connect Beukers' and Vasilyev's integrals to Sorokin's. Then, in a joint work with Rivoal, we write a very general hypergeometric series as the unique solution of a Padé approximant problem. We prove in this way that among the three numbers $\Li_s(1/2)+\frac(\log(1/2)^s)((s-1)!)$, $s \in $$2,3,4$$$, at least one is irrational. The second part is about transcendence in algebraic groups. First, we prove for some varieties a conjecture of Roy which is equivalent to the algebraic independence conjecture of logarithms of algebraic numbers. Then we prove an interpolation lemma in a commutative algebraic group $G$, which is a generalization (with multiplicities) of Masser's. When $G$ is linear, this lemma (and Fourier-Borel duality) can be stated in terms of Hopf algebras.
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Theses
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https://tel.archives-ouvertes.fr/tel-00002988
Contributor : Stephane Fischler Connect in order to contact the contributor
Submitted on : Wednesday, June 11, 2003 - 6:56:48 PM
Last modification on : Wednesday, November 17, 2021 - 12:27:02 PM
Long-term archiving on: : Friday, April 2, 2010 - 6:31:54 PM

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• HAL Id : tel-00002988, version 1

### Citation

Stéphane Fischler. Contributions à l'étude diophantienne des polylogarithmes et des groupes algébriques. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2003. Français. ⟨tel-00002988⟩

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