{"title":"A $1$-dimensional formal group over the prismatization of $\\operatorname{Spf}\\:\\mathbb{Z}_p$","authors":"Vladimir Drinfeld","doi":"10.4310/pamq.2024.v20.n1.a7","DOIUrl":null,"url":null,"abstract":"Let $\\Sigma$ denote the prismatization of $\\operatorname{Spf}\\:\\mathbb{Z}_p$. The multiplicative group over $\\Sigma$ maps to the prismatization of $\\mathbb{G}_m \\times \\operatorname{Spf}\\:\\mathbb{Z}_p$. We prove that the kernel of this map is the Cartier dual of some $1$-dimensional formal group over $\\Sigma$. We obtain some results about this formal group (e.g., we describe its Lie algebra). We give a very explicit description of the pullback of the formal group to the quotient stack $Q/\\mathbb{Z}^\\times_p$, where $Q$ is the $q$-de Rham prism.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/pamq.2024.v20.n1.a7","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
Let $\Sigma$ denote the prismatization of $\operatorname{Spf}\:\mathbb{Z}_p$. The multiplicative group over $\Sigma$ maps to the prismatization of $\mathbb{G}_m \times \operatorname{Spf}\:\mathbb{Z}_p$. We prove that the kernel of this map is the Cartier dual of some $1$-dimensional formal group over $\Sigma$. We obtain some results about this formal group (e.g., we describe its Lie algebra). We give a very explicit description of the pullback of the formal group to the quotient stack $Q/\mathbb{Z}^\times_p$, where $Q$ is the $q$-de Rham prism.