Classification of the linearly reductive finite subgroup schemes of $SL_2$

TL;DR

This paper classifies linearly reductive finite subgroup schemes of SL(2) over algebraically closed fields of positive characteristic, establishing a correspondence with certain Gorenstein local rings, thus advancing understanding of their structure.

Contribution

It provides a complete classification of these subgroup schemes and links them to isomorphism classes of specific Gorenstein rings, a novel connection in the field.

Findings

Classified all such subgroup schemes up to conjugation.

Established a one-to-one correspondence with Gorenstein local rings.

Enhanced understanding of the structure of reductive subgroup schemes.

Abstract

We classify the linearly reductive finite subgroup schemes $G$ of $SL_2=SL(V)$ over an algebraically closed field $k$ of positive characteristic, up to conjugation. As a corollary, we prove that such $G$ is in one-to-one correspondence with an isomorphism class of two-dimensional $F$-rational Gorenstein complete local rings with the coefficient field $k$ by the correspondence $G\mapsto ((\mathop{\mathrm{Sym}} V)^G)\,\hat{~}$.