{"title":"二次型与属理论:与二阶后裔的联系以及理想类的非琐特殊化应用","authors":"William Dallaporta","doi":"10.1112/jlms.12921","DOIUrl":null,"url":null,"abstract":"<p>Genus Theory is a classical feature of integral binary quadratic forms. Using the author's generalization of the well-known correspondence between quadratic form classes and ideal classes of quadratic algebras, we extend it to the case when quadratic forms are twisted and have coefficients in any principal ideal domain (PID) <span></span><math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math>. When <span></span><math>\n <semantics>\n <mrow>\n <mi>R</mi>\n <mo>=</mo>\n <mi>K</mi>\n <mo>[</mo>\n <mi>X</mi>\n <mo>]</mo>\n </mrow>\n <annotation>${R = \\mathbb {K}[X]}$</annotation>\n </semantics></math>, we show that the Genus Theory map is the quadratic form version of the 2-descent map on a certain hyperelliptic curve. As an application, we make a contribution to a question of Agboola and Pappas regarding a specialization problem of divisor classes on hyperelliptic curves. Under suitable assumptions, we prove that the set of nontrivial specializations has density 1.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12921","citationCount":"0","resultStr":"{\"title\":\"Quadratic forms and Genus Theory: A link with 2-descent and an application to nontrivial specializations of ideal classes\",\"authors\":\"William Dallaporta\",\"doi\":\"10.1112/jlms.12921\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Genus Theory is a classical feature of integral binary quadratic forms. Using the author's generalization of the well-known correspondence between quadratic form classes and ideal classes of quadratic algebras, we extend it to the case when quadratic forms are twisted and have coefficients in any principal ideal domain (PID) <span></span><math>\\n <semantics>\\n <mi>R</mi>\\n <annotation>$R$</annotation>\\n </semantics></math>. When <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>R</mi>\\n <mo>=</mo>\\n <mi>K</mi>\\n <mo>[</mo>\\n <mi>X</mi>\\n <mo>]</mo>\\n </mrow>\\n <annotation>${R = \\\\mathbb {K}[X]}$</annotation>\\n </semantics></math>, we show that the Genus Theory map is the quadratic form version of the 2-descent map on a certain hyperelliptic curve. As an application, we make a contribution to a question of Agboola and Pappas regarding a specialization problem of divisor classes on hyperelliptic curves. Under suitable assumptions, we prove that the set of nontrivial specializations has density 1.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-05-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12921\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12921\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12921","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
属理论是积分二元二次型的一个经典特征。利用作者对二次形式类与二次代数理想类之间著名对应关系的概括,我们将其扩展到二次形式是扭曲的并且在任何主理想域(PID)R $R$ 中都有系数的情况。当 R = K [ X ] ${R = \mathbb {K}[X]}$ 时,我们证明了源论映射是某个超椭圆曲线上 2-descent 映射的二次形式版本。作为应用,我们对阿格博拉和帕帕斯提出的关于超椭圆曲线上除数类的特殊化问题做出了贡献。在适当的假设条件下,我们证明了非小特化集合的密度为 1。
Quadratic forms and Genus Theory: A link with 2-descent and an application to nontrivial specializations of ideal classes
Genus Theory is a classical feature of integral binary quadratic forms. Using the author's generalization of the well-known correspondence between quadratic form classes and ideal classes of quadratic algebras, we extend it to the case when quadratic forms are twisted and have coefficients in any principal ideal domain (PID) . When , we show that the Genus Theory map is the quadratic form version of the 2-descent map on a certain hyperelliptic curve. As an application, we make a contribution to a question of Agboola and Pappas regarding a specialization problem of divisor classes on hyperelliptic curves. Under suitable assumptions, we prove that the set of nontrivial specializations has density 1.
期刊介绍:
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.