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Differences between Galois representations in automorphism and outer-automorphism groups of the fundamental group of curves

Matsumoto, M (Hiroshima)
Friday 28 August 2009, 15:30-16:30

Seminar Room 1, Newton Institute


Fix a prime l. Let C be proper smooth geometrically connected curve over a number field K, and x be its L-rational point. Let Pi denotes the pro-l completion of the geometric fundamental group of C with geometric base point over x. We have two non-abelian Galois representations: rho_A : Gal_L -> Aut(Pi) rho_O : Gal_K -> Out(Pi). Ker(rho_A) is included in Ker(rho_O). Our question is whether they differ or not: more precisely, whether or not Ker(rho_A) = (Ker(rho_O) "intersection" Gal_L.) We show that, the equality does not hold in general, by showing: Theorem: Assume that g >=3, l divides 2g-2. Then, there are infinitely many pairs (C,K) with the following property. For any extension field L with [L:K] coprime to l, and for any x in C(L), the nonequality Ker(rho_A) "not equal to" (Ker(rho_O) "intersection" Gal_L) holds. This is in contrast to the fact that for the projective line minus three point and its canonical tangential base points, the equality holds (a result of Deligne and Ihara).


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