Étale fundamental groups of smooth arithmetic surfaces and the Grothendieck conjecture

2025/11/10 by Ryoji Shimizu, Shimizu, Ryoji, Naganori Yamaguchi +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2511.06725

openalex publication_date 2025/11/10 · openalex created_date 2025/11/12 · openalex updated_date 2026/07/28

Abstract

We study the structure of the étale 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 étale fundamental group. Moreover, we obtain a partial result toward the semi-absolute version of Grothendieck's anabelian conjecture in this context.

Citations

• Quasi-projective and formal-analytic arithmetic surfaces
• Isomorphisms of Galois groups of number fields with restricted ramification
• The Neukirch-Uchida theorem with restricted ramification
• Corps de nombres peu ramifies et formes automorphes autoduales

Related

• A refined version of Grothendieck's anabelian conjecture for hyperbolic curves over finite fields
• Resolution of non-singularities and the absolute anabelian conjecture
• Etale descent obstruction and anabelian geometry of curves over finite fields
• Anabelian properties of tame fundamental groups of singular curves
• Good Sections of Arithmetic Fundamental Groups