Nicolas Daans, Vitezslav Kala, Siu Hang Man, Martin Widmer, Pavlo Yatsyna
{"title":"Most totally real fields do not have universal forms or Northcott property","authors":"Nicolas Daans, Vitezslav Kala, Siu Hang Man, Martin Widmer, Pavlo Yatsyna","doi":"arxiv-2409.11082","DOIUrl":null,"url":null,"abstract":"We show that, in the space of all totally real fields equipped with the\nconstructible topology, the set of fields that admit a universal quadratic\nform, or have the Northcott property, is meager. The main tool is a new theorem\non the number of square classes of totally positive units represented by a\nquadratic lattice of a given rank.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"40 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Number Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.11082","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 show that, in the space of all totally real fields equipped with the
constructible topology, the set of fields that admit a universal quadratic
form, or have the Northcott property, is meager. The main tool is a new theorem
on the number of square classes of totally positive units represented by a
quadratic lattice of a given rank.