Differences between Galois representations in automorphism and outer-automorphism groups of the fundamental group of curves
Seminar Room 1, Newton Institute
AbstractFix 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).
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible.