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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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