Infinitesimal commutative unipotent group schemes with one-dimensional Lie algebra

TL;DR

This paper classifies all infinitesimal commutative unipotent group schemes with one-dimensional Lie algebra over algebraically closed fields of characteristic p, providing explicit descriptions and applications to elliptic curves and group actions.

Contribution

It explicitly classifies such group schemes of order p^n, describes their structure, and applies results to subgroup schemes of supersingular elliptic curves and actions on curves.

Findings

Exactly n such group schemes up to isomorphism

Explicit descriptions of all infinitesimal subgroup schemes of supersingular elliptic curves

Answered Brion's question on rational actions of these group schemes on curves

Abstract

We prove that over an algebraically closed field of characteristic $p>0$ there are exactly, up to isomorphism, $n$ infinitesimal commutative unipotent $k$-group schemes of order $p^n$ with one-dimensional Lie algebra, and we explicitly describe them. We consequently obtain an explicit description of all infinitesimal subgroup schemes of any supersingular elliptic curve over an algebraically closed field, recovering all their $p^n$-torsions as well. Finally, we use these results to answer a question of Brion on rational actions of infinitesimal commutative unipotent group schemes on curves.