Étale Fundamental Groups of Smooth Arithmetic Surfaces and the Grothendieck Conjecture

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.