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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
Published electronically: December 1, 2010
MathSciNet review: 2748415
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Abstract: Let $ C$ be a proper smooth geometrically connected curve over a number field of $ K$ of genus $ g\geq 3$. For a fixed $ \ell$, let $ \Pi^{\ell}$ denote the pro-$ \ell$ completion of the geometric fundamental group of $ C$. For an $ L$-rational point $ x$ of $ C$, we have $ \rho_{A,x}:G_L \to \operatorname{Aut} \Pi^\ell$ associated to the base point $ x$, and its quotient by the inner automorphism group $ \rho_O:G_L \to \operatorname{Out} \pi^\ell:=\operatorname{Aut}/\operatorname{Inn}$, which is independent of the choice of $ x$. We consider whether the equality $ \operatorname{Ker} \rho_{A,x}=\operatorname{Ker} \rho_{O,x}$ 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 $ \ell$ dividing $ 2g-2$. Then there are infinitely many curves of genus $ g$ such that for any $ L$-rational point $ x$ with $ [L:K]$ finite and coprime to $ \ell$, the index $ [\operatorname{Ker} \rho_{O,x}:\operatorname{Ker} \rho_{A,x}]$ is infinite.

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Additional Information

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

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

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