Etale descent obstruction and anabelian geometry of curves over finite fields

TL;DR

This paper explores the relationship between étale fundamental groups, anabelian geometry, and the arithmetic of curves over finite fields, providing evidence for the anabelian conjecture through descent obstructions.

Contribution

It establishes a bijection between fundamental group morphisms and adelic points that survive étale descent, linking anabelian geometry to arithmetic over global function fields.

Findings

Bijection between fundamental group morphisms and adelic points

Connection of étale descent obstructions to anabelian conjecture

Evidence supporting the conjecture by relating to recent arithmetic conjectures

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 \'etale fundamental groups. The anabelian philosophy proposed by Grothendieck suggests that, when $C$ has genus at least $2$, all open homomorphisms between the \'etale 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 \'etale 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.