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Braid monodromies on proper curves and pro-ℓ Galois representations

Part of: Curves

Published online by Cambridge University Press:  01 February 2011

Naotake Takao
Naotake Takao
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan, [email protected]
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Abstract

Let C be a proper smooth geometrically connected hyperbolic curve over a field of characteristic 0 and ℓ a prime number. We prove the injectivity of the homomorphism from the pro-ℓ mapping class group attached to the two dimensional configuration space of C to the one attached to C, induced by the natural projection. We also prove a certain graded Lie algebra version of this injectivity. Consequently, we show that the kernel of the outer Galois representation on the pro-ℓ pure braid group on C with n strings does not depend on n, even if n = 1. This extends a previous result by Ihara–Kaneko. By applying these results to the universal family over the moduli space of curves, we solve completely Oda's problem on the independency of certain towers of (infinite) algebraic number fields, which has been studied by Ihara, Matsumoto, Nakamura, Ueno and the author. Sequentially we obtain certain information of the image of this Galois representation and get obstructions to the surjectivity of the Johnson–Morita homomorphism at each sufficiently large even degree (as Oda predicts), for the first time for a proper curve.

Keywords

proper hyperbolic curvebraid groupmapping class groupLie algebrapro-ℓ Galois representationuniversal monodromy representation

MSC classification

Secondary: 14H30: Coverings, fundamental group
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 11 , Issue 1 , January 2012 , pp. 161 - 188
Copyright
Copyright © Cambridge University Press 2011

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References

1.Anderson, G. W.AND Ihara, Y., Pro-ℓ branched coverings of ℙ1 and higher circular ℓ-units, Annals Math. 128 (1988),271–293.CrossRefGoogle Scholar
2.Asada, M., Two properties of the filtration of the outer automorphism groups of certain groups, Math. Z. 218 (1995), 122–133.CrossRefGoogle Scholar
3.Asada, M., On the filtration of topological and pro-ℓ mapping class groups of punctured Riemann surfaces, J. Math. Soc. Jpn 48(1) (1996), 13–36.CrossRefGoogle Scholar
4.Asada, M. and Kaneko, M., On the automorphism groups of some pro-ℓ fundamental groups, Adv. Stud. Pure Math. 12 (1987), 137–159.CrossRefGoogle Scholar
5.Asada, M. and Nakamura, H., On graded quotient modules of mapping class groups of surfaces, Israel J. Math. 90 (1995), 93–113.CrossRefGoogle Scholar
6.Bourbaki, N., Lie groups and Lie algebras (English translation), Chapter 2 (Hermann, Paris, 1976).Google Scholar
7.Cohen, F. R. and Prassidis, S., On injective homomorphisms for pure braid groups and associated Lie algebras, J. Alg. 298 (2006), 363–370.CrossRefGoogle Scholar
8.Grothendieck, A. and Raynaud, M., Revêtement étale et groupe fondamental (SGA1), Lecture Notes in Mathematics, Volume 224 (Springer, 1971).CrossRefGoogle Scholar
9.Hain, R. and Matsumoto, M., Weighted completion of Galois groups and Galois actions on the fundamental group of ℙ1 – {0, 1, ∞}, Compositio Math. 139 (2003), 119–167.CrossRefGoogle Scholar
10.Ihara, Y., The Galois representation arising from P1 – {0, 1, ∞} and Tate twists of even degree, in Galois groups over ℚ(ed. Ihara, Y., Ribet, K. and Serre, J.-P.), Mathematical Sciences Research Institute Publications, Volume 16, pp. 299–313 (Springer, 1989).CrossRefGoogle Scholar
11.Ihara, Y., Automorphisms of pure sphere braid groups and Galois representations, in The Grothendieck Festschrift, Volume II, Progress in Mathematics, Volume 87, pp. 353–373 (Birkhäuser, 1991).Google Scholar
12.Ihara, Y., Braids, Galois groups, and some arithmetic functions, in Proc. ICM90(I), pp. 99–120 (Mathematical Society of Japan, Tokyo, 1991).Google Scholar
13.Ihara, Y., On the stable derivation algebra associated to some braid groups, Israel J. Math. 80 (1992), 135–153.CrossRefGoogle Scholar
14.Ihara, Y., Some arithmetic aspects on Galois actions on the pro-p fundamental group of P1 – {0, 1, ∞}, in Arithmetic Fundamental Groups, PSPUM (ed. Fried, M. and Ihara, Y.), Volume 70, pp. 247–273 (American Mathematical Society, Providence, RI, 2002).Google Scholar
15.Ihara, Y. and Kaneko, M., Pro-ℓ pure braid groups of Riemann surfaces and Galois representations, Osaka J. Math. 29 (1992), 1–19.Google Scholar
16.Ihara, Y. and Nakamura, H., On deformation of maximally degenerate stable marked curves and Oda's problem, J. Reine Angew. Math. 487 (1997), 125–151.Google Scholar
17.Johnson, D., An abelian quotient of the mapping class group Ig, Math. Annalen 249 (1980), 225–242.CrossRefGoogle Scholar
18.Kaneko, M., Certain automorphism groups of pro-l fundamental groups of punctured Riemann surfaces, J. Fac. Sci. Univ. Tokyo 36 (1989), 363–372.Google Scholar
19.Matsumoto, M., On the galois image in the derivation algebra of π1 of the projective line minus three points, in Recent developments in inverse Galois problem, Contemporary Mathematics, Volume 186, pp. 201–213 (American Mathematical Society, Providence, RI, 1996).CrossRefGoogle Scholar
20.Matsumoto, M., Galois representations on profinite braid groups on curves, J. Reine Angew. Math. 474 (1996), 169–219.Google Scholar
21.Morita, S., Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993), 699–726.CrossRefGoogle Scholar
22.Nakamura, H., Coupling of universal monodromy representations of Galois–Teichmüller modular groups, Math. Annalen 304 (1996), 99–119.CrossRefGoogle Scholar
23.Nakamura, H., Galois rigidity of profinite fundamental groups, Sugaku Expos. 10(2) (1997), 195–215.Google Scholar
24.Nakamura, H. and Takao, N., Galois rigidity of pro-ℓ pure braid groups of algebraic curves, Trans. Am. Math. Soc. 350 (1998), 1079–1102.Google Scholar
25.Nakamura, H., Takao, N. and Ueno, R., Some stability properties of Teichmüller modular function fields with pro-l weight structures, Math. Annalen 302 (1995), 197–213.Google Scholar
26.Nakamura, H. and Tsunogai, H., Some finiteness theorems on Galois centralizers in pro-l mapping class groups, J. Reine Angew. Math. 441 (1993), 115–144.Google Scholar
27.Oda, T., A lower bound for the graded modules associated with the relative weight filtration on Teichmüller group, preprint (1992).Google Scholar
28.Oda, T., The universal monodromy representations on the pro-nilpotent fundamental groups of algebraic curves, Mathematische Arbeitstagung (Neue Serie), 9–15 Juin 1993, Max-Planck-Institute preprint MPI/93–57 (1993).Google Scholar
29.Oda, T., Etale homotopy type of the moduli spaces of algebraic curves, in Geometric Galois actions, I, London Mathematical Society Lecture Notes, Volume 242, pp. 85–95 (Cambridge University Press, 1997).CrossRefGoogle Scholar
30.Scott, G. P., Braid groups and the group of homeomorphisms of a surface, Proc. Camb. Phil. Soc. 68 (1970), 605–617.CrossRefGoogle Scholar
31.Shirshov, A. I., Subalgebras of free Lie algebras, Mat. Sb. 33 (1953), 441–452 (in Russian).Google Scholar
32.Tsunogai, H., The stable derivation algebras for higher genera, Israel J. Math. 136 (2003), 221–250.CrossRefGoogle Scholar