Etale descent obstruction and anabelian geometry of curves over finite fields

Abstract

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

Cited by in corpus (1)

• The Brauer-Manin obstruction for nonisotrivial curves over global function fields