{"title":"Infinitesimal commutative unipotent group schemes with one-dimensional Lie algebra","authors":"Bianca Gouthier","doi":"arxiv-2409.11997","DOIUrl":null,"url":null,"abstract":"We prove that over an algebraically closed field of characteristic $p>0$\nthere are exactly, up to isomorphism, $n$ infinitesimal commutative unipotent\n$k$-group schemes of order $p^n$ with one-dimensional Lie algebra, and we\nexplicitly describe them. We consequently recover an explicit description of\nthe $p^n$-torsion of any supersingular elliptic curve over an algebraically\nclosed field. Finally, we use these results to answer a question of Brion on\nrational actions of infinitesimal commutative unipotent group schemes on\ncurves.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"66 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Algebraic Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.11997","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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 recover an explicit description of
the $p^n$-torsion of any supersingular elliptic curve over an algebraically
closed field. Finally, we use these results to answer a question of Brion on
rational actions of infinitesimal commutative unipotent group schemes on
curves.