On the Specialization Homomorphism of Fundamental Groups of Curves in Positive Characteristic

Introduction Recall that for proper smooth and connected curves of genus g ≥ 2 over an algebraically closed eld of characteristic 0 the structure of the étale fundamental group π g is well known and depends only on the genus g. Namely it is the pro-nite completion of the topological fundamental group of a compact orientable topological surface of genus g. In contrast to this, the structure of the étale fundamental group of proper smooth and connected curves of genus g ≥ 2 in positive characteristic is unknown, and it depends on the isomorphy type of the curve in discussion. The aim of this paper is to give new evidence for anabelian phenomena for proper curves over algebraically closed elds of characteristic p > 0. Before going into the details of the results we are going to prove, we set some notation and recall well known facts. Let k be an algebraically closed eld of characteristic p > 0. Let X be a projective smooth and connected curve of genus g ≥ 2 over k, and let J be the Jacobian of X. We denote by π 1 (X), π p 1 (X), and π p 1 (X) the étale fundamental group of X, its prop quotient, and its prime to p quotient. Then: (1) The structure of π p 1 (X) is given by Shafarevich's Theorem; see [Sh]. It is isomorphic to the prop free group on r := r X generators, where r X is the prank of J. (2) The structure of π p 1 (X) is well known by Grothendieck's Specialization Theorem ; see [SGA-1]. It is the prime to p completion of the topological fundamental group of a compact orientable topological surface of genus g. (3) In contrast to this, the structure of the whole fundamental group π 1 (X) is a big mystery! Its structure is not known in any single case. However, by Grothendieck's Specialization Theorem we know that π 1 (X) is the quotient of 107 108 FLORIAN POP AND MOHAMED SAïDI the pronite completion Π g of the topological fundamental group of a compact orientable topological surface of genus g. In particular π 1 (X) is topologically nitely generated. Since such groups are completely determined by the set of their nite quotients, another interpretation of (1) is the following: If two curves as above have the same prank , then there …