Étale Fundamental Groups of Smooth Arithmetic Surfaces and the Grothendieck Conjecture
Ryoji Shimizu, Naganori Yamaguchi
TL;DR
This work extends Grothendieck's anabelian program to relative and semi-absolute settings for hyperbolic curves over arithmetic bases. It proves a relative Grothendieck conjecture for hyperbolic curves over schemes with an invertible rational prime on the base, and exhibits counterexamples when no such prime exists. It then develops a semi-absolute theory over rings of S-integers, reconstructing genus, cusps, and the p-adic cyclotomic character from outer Galois representations and establishing isomorphism-detection results under large Dirichlet densities. Finally, it demonstrates how the geometric generic fiber can be recovered from profinite data, leveraging weight theory, inertia/decomposition analysis, and p-adic Mochizuki–Tsujimura-type results, thereby advancing the reach of anabelian geometry in arithmetic-surface contexts.
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.