Etale descent obstruction and anabelian geometry of curves over finite fields

2023/06/08 by Brendan Creutz, Jose Felipe Voloch, José Felipe Voloch
Mathematics · #math.NT #math.AG

paper · pdf · doi:10.46298/epiga.2024.11483

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.

Related

• Étale fundamental groups of smooth arithmetic surfaces and the Grothendieck conjecture
• Resolution of non-singularities and the absolute anabelian conjecture
• A refined version of Grothendieck's anabelian conjecture for hyperbolic curves over finite fields
• Anabelian properties of tame fundamental groups of singular curves
• Profinite complexes of curves, their automorphisms and anabelian properties of moduli stacks of curves