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

Summary

Introduction

Recall that for proper smooth and connected curves of genus g ≥ 2 over an algebraically closed field of characteristic 0 the structure of the étale fundamental groupis well known and depends only on the genus g. Namely it is the profinite 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 fields 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 field 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), , and the étale fundamental group of X, its pro-p quotient, and its prime to p quotient. Then:

(1) The structure of π ₁ ^p(X) is given by Shafarevich's Theorem; see [Sh]. It is isomorphic to the pro-p free group on r := rx generators, where rx is the p-rank of J.