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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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  • 1. Michael P. Anderson, Exactness properties of profinite completion functors, Topology 13 (1974), 229–239. MR 0354882
  • 2. P. Deligne, Le groupe fondamental de la droite projective moins trois points, Galois groups over 𝑄 (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 79–297 (French). MR 1012168, 10.1007/978-1-4613-9649-9_3
  • 3. Revêtements étales et groupe fondamental, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1); Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud; Lecture Notes in Mathematics, Vol. 224. MR 0354651
  • 4. Richard Hain and David Reed, Geometric proofs of some results of Morita, J. Algebraic Geom. 10 (2001), no. 2, 199–217. MR 1811554
  • 5. Yasutaka Ihara, Profinite braid groups, Galois representations and complex multiplications, Ann. of Math. (2) 123 (1986), no. 1, 43–106. MR 825839, 10.2307/1971352
  • 6. Yasutaka Ihara, Braids, Galois groups, and some arithmetic functions, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 99–120. MR 1159208
  • 7. John P. Labute, On the descending central series of groups with a single defining relation, J. Algebra 14 (1970), 16–23. MR 0251111
  • 8. Makoto Matsumoto and Akio Tamagawa, Mapping-class-group action versus Galois action on profinite fundamental groups, Amer. J. Math. 122 (2000), no. 5, 1017–1026. MR 1781929
  • 9. Shigeyuki Morita, Families of Jacobian manifolds and characteristic classes of surface bundles. I, Ann. Inst. Fourier (Grenoble) 39 (1989), no. 3, 777–810 (English, with French summary). MR 1030850
  • 10. Takayuki Oda, Etale homotopy type of the moduli spaces of algebraic curves, Geometric Galois actions, 1, London Math. Soc. Lecture Note Ser., vol. 242, Cambridge Univ. Press, Cambridge, 1997, pp. 85–95. MR 1483111

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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