Difference between Galois representations in automorphism and outer-automorphism groups of a fundamental group

Author:
Makoto Matsumoto

Journal:
Proc. Amer. Math. Soc. **139** (2011), 1215-1220

MSC (2010):
Primary 11R32; Secondary 14H30, 14H10, 20J05

DOI:
https://doi.org/10.1090/S0002-9939-2010-10846-8

Published electronically:
December 1, 2010

MathSciNet review:
2748415

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a proper smooth geometrically connected curve over a number field of of genus . For a fixed , let denote the pro- completion of the geometric fundamental group of . For an -rational point of , we have associated to the base point , and its quotient by the inner automorphism group , which is independent of the choice of . We consider whether the equality holds or not. Deligne and Ihara showed the equality when the curve is the projective line minus three points with a choice of tangential basepoint. The result here is: Fix an dividing . Then there are infinitely many curves of genus such that for any -rational point with finite and coprime to , the index is infinite.

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Additional Information

**Makoto Matsumoto**

Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, Tokyo, 153-8914 Japan

Email:
matumoto@ms.u-tokyo.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-2010-10846-8

Received by editor(s):
September 2, 2009

Received by editor(s) in revised form:
April 19, 2010

Published electronically:
December 1, 2010

Additional Notes:
The author was supported in part by the Scientific Grants-in-Aid 16204002 and 19204002 and by the Core-to-Core grant 18005 from the Japan Society for the Promotion of Science. Part of this study was done while the author visited the Newton Institute in August 2009.

Dedicated:
Dedicated to Professor Takayuki Oda on his 60th birthday

Communicated by:
Ted Chinburg

Article copyright:
© Copyright 2010
American Mathematical Society