{"title":"论两个圆锥的乘积的还原无ramified 维特群","authors":"Alexander S. Sivatski","doi":"10.1007/s00229-024-01591-x","DOIUrl":null,"url":null,"abstract":"<p>We investigate the reduced unramified Witt group of the product of two smooth projective conics <span>\\(X_1\\)</span>, <span>\\(X_2\\)</span> over a field. In some particular cases, this group denoted as <span>\\(W(X_1,X_2)\\)</span> turns out to be very small (zero or <span>\\({\\mathbb {Z}}/2{\\mathbb {Z}}\\)</span>). On the other hand, certain examples when it is infinite are constructed. We give sufficient conditions providing nontriviality of <span>\\(W(X_1,X_2)\\)</span> in terms of 2-fold Pfister forms <span>\\(\\pi _1\\)</span>, <span>\\(\\pi _2\\)</span> associated with the conics. These conditions and constructions of the corresponding nonzero elements in <span>\\(W(X_1,X_2)\\)</span> depend on <span>\\({\\text {ind}}(\\pi _1+\\pi _2)\\)</span>. Also we study the question of triviality (nontriviality) of this group with respect to extensions of the ground field.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the reduced unramified Witt group of the product of two conics\",\"authors\":\"Alexander S. Sivatski\",\"doi\":\"10.1007/s00229-024-01591-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We investigate the reduced unramified Witt group of the product of two smooth projective conics <span>\\\\(X_1\\\\)</span>, <span>\\\\(X_2\\\\)</span> over a field. In some particular cases, this group denoted as <span>\\\\(W(X_1,X_2)\\\\)</span> turns out to be very small (zero or <span>\\\\({\\\\mathbb {Z}}/2{\\\\mathbb {Z}}\\\\)</span>). On the other hand, certain examples when it is infinite are constructed. We give sufficient conditions providing nontriviality of <span>\\\\(W(X_1,X_2)\\\\)</span> in terms of 2-fold Pfister forms <span>\\\\(\\\\pi _1\\\\)</span>, <span>\\\\(\\\\pi _2\\\\)</span> associated with the conics. These conditions and constructions of the corresponding nonzero elements in <span>\\\\(W(X_1,X_2)\\\\)</span> depend on <span>\\\\({\\\\text {ind}}(\\\\pi _1+\\\\pi _2)\\\\)</span>. Also we study the question of triviality (nontriviality) of this group with respect to extensions of the ground field.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-09-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00229-024-01591-x\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00229-024-01591-x","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On the reduced unramified Witt group of the product of two conics
We investigate the reduced unramified Witt group of the product of two smooth projective conics \(X_1\), \(X_2\) over a field. In some particular cases, this group denoted as \(W(X_1,X_2)\) turns out to be very small (zero or \({\mathbb {Z}}/2{\mathbb {Z}}\)). On the other hand, certain examples when it is infinite are constructed. We give sufficient conditions providing nontriviality of \(W(X_1,X_2)\) in terms of 2-fold Pfister forms \(\pi _1\), \(\pi _2\) associated with the conics. These conditions and constructions of the corresponding nonzero elements in \(W(X_1,X_2)\) depend on \({\text {ind}}(\pi _1+\pi _2)\). Also we study the question of triviality (nontriviality) of this group with respect to extensions of the ground field.