Etale descent obstruction and anabelian geometry of curves over fi nite fi elds

Let C and D be smooth, proper and geometrically integral curves over a fi nite fi eld F . Any morphism D- C induces a morphism of & eacute;tale fundamental groups pi 1 (D)- pi 1 (C) . The anabelian philosophy proposed by Grothendieck suggests that, when C has genus at least 2 , all open homomorphisms between the & eacute;tale fundamental groups should arise in this way from a nonconstant morphism of curves. We relate this expectation to the arithmetic of the curve C considered as a curve over the global function fi eld K = F (D) . Speci fi cally, we show that there is a bijection between the set of conjugacy classes of well-behaved morphisms of fundamental groups and locally constant adelic points of C that survive & eacute;tale descent. We use this to provide further evidence for the anabelian conjecture and relate it to another recent conjecture by Sutherland and the second author.