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Weighted Completion of Galois Groups and Galois Actions on the Fundamental Group of P1 − {0,1, ∞}

Published online by Cambridge University Press:  04 December 2007

Richard Hain  and
Makoto Matsumoto
Richard Hain
Affiliation:
Department of Mathematics, Duke University, Box 90320, Durham, NC 27708-0320, U.S.A. e-mail: [email protected]
Makoto Matsumoto
Affiliation:
Department of Mathematics, Faculty of Science, Hiroshima University, Higashi-Hiroshima, 739-8526 Japan. e-mail: [email protected]
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Abstract

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Fix a prime number [ell ]. We prove a conjecture stated by Ihara, which he attributes to Deligne, about the action of the absolute Galois group on the pro-[ell ] completion of the fundamental group of the thrice punctured projective line. Similar techniques are also used to prove part of a conjecture of Goncharov, also about the action of the absolute Galois group on the fundamental group of the thrice punctured projective line. The main technical tool is the weighted completion of a profinite group with respect to a reductive representation (and other appropriate data).

Keywords

Galois cohomologyGalois groupmixed Tate motiveTannakian categoryweighted completion
Type
Research Article
Information
Compositio Mathematica , Volume 139 , Issue 2 , November 2003 , pp. 119 - 167
Copyright
© 2003 Kluwer Academic Publishers