{"title":"$SL_2$和Markoff曲面上的换向子1","authors":"Amit Ghosh, C. Meiri, P. Sarnak","doi":"10.53733/198","DOIUrl":null,"url":null,"abstract":"We show that the commutator equation over $SL_2\\Z$ satisfies a profinite local to global principle, while it can fail with infinitely many exceptions for $SL_2(\\Z[\\frac{1}{p}])$. The source of the failure is a reciprocity obstruction to the Hasse Principle for cubic Markoff surfaces.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"10 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2021-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Commutators in $SL_2$ and Markoff surfaces I\",\"authors\":\"Amit Ghosh, C. Meiri, P. Sarnak\",\"doi\":\"10.53733/198\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We show that the commutator equation over $SL_2\\\\Z$ satisfies a profinite local to global principle, while it can fail with infinitely many exceptions for $SL_2(\\\\Z[\\\\frac{1}{p}])$. The source of the failure is a reciprocity obstruction to the Hasse Principle for cubic Markoff surfaces.\",\"PeriodicalId\":30137,\"journal\":{\"name\":\"New Zealand Journal of Mathematics\",\"volume\":\"10 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-10-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"New Zealand Journal of Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.53733/198\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"New Zealand Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.53733/198","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}
We show that the commutator equation over $SL_2\Z$ satisfies a profinite local to global principle, while it can fail with infinitely many exceptions for $SL_2(\Z[\frac{1}{p}])$. The source of the failure is a reciprocity obstruction to the Hasse Principle for cubic Markoff surfaces.
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