Appears in collection : Rational points on irrational varieties

Let C and D be proper geometrically integral curves over a finite field and let K be the function field of D. I will discuss descent obstructions to the existence of K-rational points on C. In this context, an adelic point on C can be viewed as a Galois equivariant map between the geometric points of D and C. We show that the adelic points surviving abelian descent correspond to those maps which extend to homomorphisms of the Jacobians and use this to prove that abelian descent cuts out the set of rational points when the genus of D is less than the genus of C. I will also present a connection between the finite etale descent obstruction and anabelian geometry. This is joint work with Felipe Voloch.