{"title":"叠曲线上积分点的局部-全局原理","authors":"M. Bhargava, B. Poonen","doi":"10.1090/jag/796","DOIUrl":null,"url":null,"abstract":"<p>We construct a stacky curve of genus <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1 slash 2\">\n <mml:semantics>\n <mml:mrow>\n <mml:mn>1</mml:mn>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mn>2</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">1/2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> (i.e., Euler characteristic <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1\">\n <mml:semantics>\n <mml:mn>1</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>) over <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Z\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {Z}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> that has an <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {R}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-point and a <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Z Subscript p\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\n </mml:mrow>\n <mml:mi>p</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {Z}_p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-point for every prime <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\n <mml:semantics>\n <mml:mi>p</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> but no <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Z\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {Z}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-point. This is best possible: we also prove that any stacky curve of genus less than <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1 slash 2\">\n <mml:semantics>\n <mml:mrow>\n <mml:mn>1</mml:mn>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mn>2</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">1/2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> over a ring of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S\">\n <mml:semantics>\n <mml:mi>S</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">S</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-integers of a global field <italic>satisfies</italic> the local-global principle for integral points.</p>","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2020-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":"{\"title\":\"The local-global principle for integral points on stacky curves\",\"authors\":\"M. Bhargava, B. Poonen\",\"doi\":\"10.1090/jag/796\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We construct a stacky curve of genus <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"1 slash 2\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mn>1</mml:mn>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo>/</mml:mo>\\n </mml:mrow>\\n <mml:mn>2</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">1/2</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> (i.e., Euler characteristic <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"1\\\">\\n <mml:semantics>\\n <mml:mn>1</mml:mn>\\n <mml:annotation encoding=\\\"application/x-tex\\\">1</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>) over <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper Z\\\">\\n <mml:semantics>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">Z</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {Z}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> that has an <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper R\\\">\\n <mml:semantics>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {R}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-point and a <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper Z Subscript p\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">Z</mml:mi>\\n </mml:mrow>\\n <mml:mi>p</mml:mi>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {Z}_p</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-point for every prime <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"p\\\">\\n <mml:semantics>\\n <mml:mi>p</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">p</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> but no <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper Z\\\">\\n <mml:semantics>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">Z</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {Z}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-point. This is best possible: we also prove that any stacky curve of genus less than <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"1 slash 2\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mn>1</mml:mn>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo>/</mml:mo>\\n </mml:mrow>\\n <mml:mn>2</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">1/2</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> over a ring of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper S\\\">\\n <mml:semantics>\\n <mml:mi>S</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">S</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-integers of a global field <italic>satisfies</italic> the local-global principle for integral points.</p>\",\"PeriodicalId\":54887,\"journal\":{\"name\":\"Journal of Algebraic Geometry\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2020-05-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"7\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Algebraic Geometry\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/jag/796\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebraic Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/jag/796","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
The local-global principle for integral points on stacky curves
We construct a stacky curve of genus 1/21/2 (i.e., Euler characteristic 11) over Z\mathbb {Z} that has an R\mathbb {R}-point and a Zp\mathbb {Z}_p-point for every prime pp but no Z\mathbb {Z}-point. This is best possible: we also prove that any stacky curve of genus less than 1/21/2 over a ring of SS-integers of a global field satisfies the local-global principle for integral points.
期刊介绍：
The Journal of Algebraic Geometry is devoted to research articles in algebraic geometry, singularity theory, and related subjects such as number theory, commutative algebra, projective geometry, complex geometry, and geometric topology.
This journal, published quarterly with articles electronically published individually before appearing in an issue, is distributed by the American Mathematical Society (AMS). In order to take advantage of some features offered for this journal, users will occasionally be linked to pages on the AMS website.
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