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The Grothendieck-Teichmüller group and Galois theory of the rational numbers – European network GTEM. (English) Zbl 1111.11050

Laptev, Ari (ed.), Proceedings of the 4th European congress of mathematics (ECM), Stockholm, Sweden, June 27–July 2, 2004. Zürich: European Mathematical Society (EMS) (ISBN 3-03719-009-4/hbk). 681-696 (2005).
In this paper, several branches of expository accounts are given on nonabelian Galois action, Grothendieck-Teichmüller theory and anabelian geomety from the activities of GTEM – European Network “Galois Theory and Explicit Method”. Researches within this network are listed as (1) (Explicit) finite Galois groups over \(\mathbb Q\). (2) The Inverse Galois Problem (IGP). (3) Dessins d’enfants. (4) Grothendieck-Teichmüller Theory: \(\roman{Gal}_{\mathbb Q}\) and \(\roman{GT}\). (5) Arithmetic of elliptic curves over number fields. (6) Algorithms in number theory, in particular class field theory. (7) Differential Galois Theory (8) Arithmetic of covers, arithmetic fundamental groups. (9) (Birational) anabelian geometry. (10) Miscellaneous: Iwasawa Theory, Invariant Theory, explicit implementations,... The author selected several topics mainly concerned with the ground field of the rational numbers, giving overviews on those subjects, positioning new results, and supplying bibliographical informations.
For the entire collection see [Zbl 1064.00004].

MSC:

11R32 Galois theory
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11G20 Curves over finite and local fields
12F12 Inverse Galois theory
14H30 Coverings of curves, fundamental group
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