\'Etale Fundamental Groups of Smooth Arithmetic Surfaces and the Grothendieck Conjecture

TL;DR

This paper explores the structure of étale fundamental groups of smooth arithmetic surfaces and advances the understanding of Grothendieck's anabelian conjecture, showing hyperbolic curves over certain number rings are fully determined by their fundamental groups.

Contribution

It proves hyperbolic curves over rings of S-integers with inverted primes are anabelian and provides partial results on a semi-absolute version of the conjecture.

Findings

Hyperbolic curves over S-integers are anabelian.

Partial progress on semi-absolute Grothendieck conjecture.

Fundamental groups determine the schematic structure of these curves.

Abstract

We study the structure of the \'etale fundamental groups of smooth curves over certain arithmetic schemes, and investigate the relative version of Grothendieck's anabelian conjecture in this setting. Consequently, every hyperbolic curve over the ring of S-integers of a number field in which a rational prime is inverted is anabelian, i.e., its schematic structure is completely determined by its \'etale fundamental group. Moreover, we obtain a partial result toward the semi-absolute version of Grothendieck's anabelian conjecture in this context.