代数闭域上曲线群的对偶定理

IF 0.4 4区数学 Q4 MATHEMATICS

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Pub Date : 2018-10-04 DOI:10.1007/s12188-018-0196-7

Timo Keller

{"title":"代数闭域上曲线群的对偶定理","authors":"Timo Keller","doi":"10.1007/s12188-018-0196-7","DOIUrl":null,"url":null,"abstract":"<div>In this note, we prove a duality theorem for the Tate–Shafarevich group of a finite discrete Galois module over the function field K of a curve over an algebraically closed field: there is a perfect duality of finite groups <img> for F a finite étale Galois module on K of order invertible in K and with \\(F' = {{\\mathrm{Hom}}}(F,\\mathbf{Q}/\\mathbf {Z}(1))\\). Furthermore, we prove that \\(\\mathrm {H}^1(K,G) = 0\\) for G a simply connected, quasisplit semisimple group over K not of type \\(E_8\\).</div>","PeriodicalId":50932,"journal":{"name":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","volume":"88 2","pages":"289 - 295"},"PeriodicalIF":0.4000,"publicationDate":"2018-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12188-018-0196-7","citationCount":"0","resultStr":"{\"title\":\"A duality theorem for Tate–Shafarevich groups of curves over algebraically closed fields\",\"authors\":\"Timo Keller\",\"doi\":\"10.1007/s12188-018-0196-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>In this note, we prove a duality theorem for the Tate–Shafarevich group of a finite discrete Galois module over the function field K of a curve over an algebraically closed field: there is a perfect duality of finite groups <img> for F a finite étale Galois module on K of order invertible in K and with \\\\(F' = {{\\\\mathrm{Hom}}}(F,\\\\mathbf{Q}/\\\\mathbf {Z}(1))\\\\). Furthermore, we prove that \\\\(\\\\mathrm {H}^1(K,G) = 0\\\\) for G a simply connected, quasisplit semisimple group over K not of type \\\\(E_8\\\\).</div>\",\"PeriodicalId\":50932,\"journal\":{\"name\":\"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg\",\"volume\":\"88 2\",\"pages\":\"289 - 295\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2018-10-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1007/s12188-018-0196-7\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s12188-018-0196-7\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s12188-018-0196-7","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

本文证明了代数闭域上曲线函数域K上有限离散伽罗瓦模的Tate-Shafarevich群的对偶定理:在K上的阶可逆的K上的有限离散伽罗瓦模存在有限群的完全对偶性 \(F' = {{\mathrm{Hom}}}(F,\mathbf{Q}/\mathbf {Z}(1))\)．进一步证明 \(\mathrm {H}^1(K,G) = 0\) 对于K非型上的一个单连通拟分裂半单群 \(E_8\)．

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

A duality theorem for Tate–Shafarevich groups of curves over algebraically closed fields

In this note, we prove a duality theorem for the Tate–Shafarevich group of a finite discrete Galois module over the function field K of a curve over an algebraically closed field: there is a perfect duality of finite groups for F a finite étale Galois module on K of order invertible in K and with \(F' = {{\mathrm{Hom}}}(F,\mathbf{Q}/\mathbf {Z}(1))\). Furthermore, we prove that \(\mathrm {H}^1(K,G) = 0\) for G a simply connected, quasisplit semisimple group over K not of type \(E_8\).

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 数学-数学

CiteScore

0.80

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： The first issue of the "Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg" was published in the year 1921. This international mathematical journal has since then provided a forum for significant research contributions. The journal covers all central areas of pure mathematics, such as algebra, complex analysis and geometry, differential geometry and global analysis, graph theory and discrete mathematics, Lie theory, number theory, and algebraic topology.