{"title":"Artin Tate和Birch Swinnerton Dyer的猜想","authors":"S. Lichtenbaum, N. Ramachandran, T. Suzuki","doi":"10.46298/epiga.2022.7482","DOIUrl":null,"url":null,"abstract":"We provide two proofs that the conjecture of Artin-Tate for a fibered surface\nis equivalent to the conjecture of Birch-Swinnerton-Dyer for the Jacobian of\nthe generic fibre. As a byproduct, we obtain a new proof of a theorem of\nGeisser relating the orders of the Brauer group and the Tate-Shafarevich group.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":" ","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"The conjectures of Artin-Tate and Birch-Swinnerton-Dyer\",\"authors\":\"S. Lichtenbaum, N. Ramachandran, T. Suzuki\",\"doi\":\"10.46298/epiga.2022.7482\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We provide two proofs that the conjecture of Artin-Tate for a fibered surface\\nis equivalent to the conjecture of Birch-Swinnerton-Dyer for the Jacobian of\\nthe generic fibre. As a byproduct, we obtain a new proof of a theorem of\\nGeisser relating the orders of the Brauer group and the Tate-Shafarevich group.\",\"PeriodicalId\":41470,\"journal\":{\"name\":\"Epijournal de Geometrie Algebrique\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2021-01-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Epijournal de Geometrie Algebrique\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.46298/epiga.2022.7482\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Epijournal de Geometrie Algebrique","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.46298/epiga.2022.7482","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
The conjectures of Artin-Tate and Birch-Swinnerton-Dyer
We provide two proofs that the conjecture of Artin-Tate for a fibered surface
is equivalent to the conjecture of Birch-Swinnerton-Dyer for the Jacobian of
the generic fibre. As a byproduct, we obtain a new proof of a theorem of
Geisser relating the orders of the Brauer group and the Tate-Shafarevich group.
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