Curves with Innnite K-rational Geometric Fundamental Group

1 Rational Geometric Fundamental Groups Let K be a nitely generated eld with separable closure K s and absolute Galois group G K : By a curve C=K we always understand a smooth geometrically irreducible projective curve. Let F(C) be its function eld and let (C) be the Galois group of the maximal unramiied extension of F(C): We have the exact sequence 1 ! g (C) ! (C) ! G K ! 1 (?) where g (C) is the geometric (proonite) fundamental group of C Spec(K s) (i.e. g (C) is equal to the Galois group of the maximal unramiied extension of F(C) K s): This sequence induces a homomorphism C from G K to Out((g (C)) which is the group of automorphisms modulo inner automorphisms of g (C). It is well known that C is an important tool for studying C. For instance, it determines C up to K?isomorphisms if the genus of C is at least 2 and K is a number eld or even a p?adic eld (see Mo]). So it is of interest to nd quotients (C) of (C) such that the induced map of G K is not the identity but the induced representation C becomes trivial. We give a geometric interpretation of such quotients. For this we assume that the sequence (?) is split and choose a section s which induces a homo-morphism from G K to Aut((g (C)). Let U be a normal subgroup of (C) contained in g (C). The representation C becomes trivial modulo U if and only if the map mod U has its image inside of Inn((g (C)=U). Let Z be the center of g (C)=U and = ((g (C)=U)=Z: Our condition on U implies that there is exactly one group theoretical section s from G K to ((C)=U)=Z inducing the trivial action on and so occurs as Galois group of an unramiied regular extension of F(C) in a natural way. Thus, to nd center free innnite factors of g on which C becomes trivial 1