Aspherical neighborhoods on arithmetic surfaces: the local case

TL;DR

This paper investigates the local geometric structure of arithmetic surfaces over local rings of integers, focusing on the existence of neighborhoods with specific homotopy types related to finite group classes.

Contribution

It introduces criteria for neighborhoods on arithmetic surfaces to have $K(\pi,1)$-type homotopy types in the context of a full class of finite groups.

Findings

Existence of neighborhoods with $K(\pi,1)$-type homotopy types under certain conditions

Characterization of local geometric points with such neighborhoods

Insights into the étale homotopy types of arithmetic surfaces

Abstract

On arithmetic surfaces over local rings of integers we examine whether any geometric point has a basis of \'etale neighborhoods whose $\mathfrak{c}$-completed \'etale homotopy types are of type $K(\pi,1)$ for a given full class~$\mathfrak{c}$ of finite groups.