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Galois theory, motives and transcendental numbers. (English) Zbl 1219.11109

Connes, Alain (ed.) et al., Renormalization and Galois theories. Selected papers of the CIRM workshop, Luminy, France, March 2006. Zürich: European Mathematical Society (ISBN 978-3-03719-073-9/pbk). IRMA Lectures in Mathematics and Theoretical Physics 15, 165-177 (2009).
The initial question of this nice little paper is whether a Galois theory of transcendental could be envisioned. The first examples the author describes are those of \(\pi\) and of elliptic periods, where he suggests a naive approach. Next he shows that this naive point of view seems to lead to a dead end, but later in the paper he explains why the answers in these two special cases are sound.
The next section deals with periods and motives. He calls “effective periods” what M. Kontsevich and D. Zagier call periods in their seminal paper [Mathematics unlimited – 2001 and beyond. Berlin: Springer. 771–808 (2001; Zbl 1039.11002)] and he introduces the algebra of periods by inverting \(2i\pi\). He gives references to the related papers [P. Belkale and P. Brosnan, “Periods and Igusa local zeta functions”, Int. Math. Res. Not. 2003, No. 49, 2655–2670 (2003; Zbl 1067.11075)] and [C. Bogner and S. Weinzierl, “Periods and Feynman integrals”, J. Math. Phys. 50, No. 4, Paper No. 042302, 16 p. (2009; Zbl 1214.81096)] and quotes also Yoshinaga who answered a question of Kontsevich and Zagier by giving examples of complex numbers which are not periods, using complexity theory [M. Yoshinaga, “Periods and elementary real numbers”, arXiv:0805.0349v1].
Next the author discusses Grothendieck’s period conjecture and its relationship with a conjecture of Kontsevich. These conjectures generalize the classical conjecture on the algebraic independence of logarithms of algebraic numbers; the author’s conjecture [Y. André, Une introduction aux motifs. Motifs purs, motifs mixtes, périodes. Panoramas et Synthèses 17. Paris: Société Mathématique de France (2004; Zbl 1060.14001)] also includes the more general conjecture of Schanuel. Then the author proposes a Galois theory of periods which extends the classical Galois theory of algebraic numbers and which reduces to the above mentioned naive approach for \(\pi\) and elliptic periods. The cases of the Euler Gamma function with the conjectures of Rohrlich and Lang, of logarithms of algebraic numbers, of zeta values and multiple zeta values, are also discussed. For MZV see also the recent paper by F. Brown [“Mixed Tate motives over \(\mathbb Z\)”, arXiv:1102.1312v1]. The last section deals with the relationship between this extended Galois theory and differential Galois theory.
For the entire collection see [Zbl 1173.14003].

MSC:

11J81 Transcendence (general theory)
12F10 Separable extensions, Galois theory
14C15 (Equivariant) Chow groups and rings; motives
14F40 de Rham cohomology and algebraic geometry
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