An arithmetic analog of Klein's classification of finite subgroups of $\mathrm{SL}_2(\mathbb{C})$

TL;DR

This paper classifies finite, flat, linearly reductive subgroup schemes of SL_2 over the spectrum of the ring of integers of a number field, extending Klein's classical classification to an arithmetic setting.

Contribution

It provides a comprehensive classification and finiteness results for such subgroup schemes over number rings, linking algebraic group theory with arithmetic geometry.

Findings

Classification of subgroup schemes over number rings

Finiteness results for these subgroup schemes

Density results for associated quotient singularities

Abstract

Let $K$ be a number field with ring of integers $\mathcal{O}_K$. We describe and classify finite, flat, and linearly reductive subgroup schemes of $\mathrm{SL}_2$ over $\mathrm{Spec}\:\mathcal{O}_K$. We also establish finiteness results for these group schemes, as well as density results for the associated quotient singularities.