{"title":"大的田地并不大","authors":"B. Mazur, K. Rubin","doi":"10.1090/bproc/57","DOIUrl":null,"url":null,"abstract":"<p>A subfield <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\n <mml:semantics>\n <mml:mi>K</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q overbar\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mover>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">¯<!-- ¯ --></mml:mo>\n </mml:mover>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\bar {\\mathbb {Q}}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is <italic>large</italic> if every smooth curve <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C\">\n <mml:semantics>\n <mml:mi>C</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">C</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=\"upper K\">\n <mml:semantics>\n <mml:mi>K</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> with a <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\n <mml:semantics>\n <mml:mi>K</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-rational point has infinitely many <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\n <mml:semantics>\n <mml:mi>K</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-rational points. A subfield <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\n <mml:semantics>\n <mml:mi>K</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q overbar\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mover>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">¯<!-- ¯ --></mml:mo>\n </mml:mover>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\bar {\\mathbb {Q}}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is <italic>big</italic> if for every positive integer <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\n <mml:semantics>\n <mml:mi>n</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\n <mml:semantics>\n <mml:mi>K</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> contains a number field <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\n <mml:semantics>\n <mml:mi>F</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> with <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-bracket upper F colon double-struck upper Q right-bracket\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">[</mml:mo>\n <mml:mi>F</mml:mi>\n <mml:mo>:</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">]</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">[F:\\mathbb {Q}]</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> divisible by <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\n <mml:semantics>\n <mml:mi>n</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. The question of whether all big fields are large seems to have circulated for some time, although we have been unable to find its origin. In this paper we show that there are big fields that are not large.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"8 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2020-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Big fields that are not large\",\"authors\":\"B. Mazur, K. Rubin\",\"doi\":\"10.1090/bproc/57\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A subfield <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper K\\\">\\n <mml:semantics>\\n <mml:mi>K</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">K</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper Q overbar\\\">\\n <mml:semantics>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mover>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">Q</mml:mi>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">¯<!-- ¯ --></mml:mo>\\n </mml:mover>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\bar {\\\\mathbb {Q}}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is <italic>large</italic> if every smooth curve <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper C\\\">\\n <mml:semantics>\\n <mml:mi>C</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">C</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=\\\"upper K\\\">\\n <mml:semantics>\\n <mml:mi>K</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">K</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> with a <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper K\\\">\\n <mml:semantics>\\n <mml:mi>K</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">K</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-rational point has infinitely many <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper K\\\">\\n <mml:semantics>\\n <mml:mi>K</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">K</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-rational points. A subfield <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper K\\\">\\n <mml:semantics>\\n <mml:mi>K</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">K</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper Q overbar\\\">\\n <mml:semantics>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mover>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">Q</mml:mi>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">¯<!-- ¯ --></mml:mo>\\n </mml:mover>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\bar {\\\\mathbb {Q}}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is <italic>big</italic> if for every positive integer <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"n\\\">\\n <mml:semantics>\\n <mml:mi>n</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">n</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper K\\\">\\n <mml:semantics>\\n <mml:mi>K</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">K</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> contains a number field <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper F\\\">\\n <mml:semantics>\\n <mml:mi>F</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">F</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> with <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-bracket upper F colon double-struck upper Q right-bracket\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">[</mml:mo>\\n <mml:mi>F</mml:mi>\\n <mml:mo>:</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">Q</mml:mi>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">]</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">[F:\\\\mathbb {Q}]</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> divisible by <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"n\\\">\\n <mml:semantics>\\n <mml:mi>n</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">n</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. The question of whether all big fields are large seems to have circulated for some time, although we have been unable to find its origin. In this paper we show that there are big fields that are not large.</p>\",\"PeriodicalId\":106316,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society, Series B\",\"volume\":\"8 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-04-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society, Series B\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/bproc/57\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/bproc/57","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 3
摘要
Q¯\bar {\mathbb {Q}}的子域K K是大的，如果每条光滑曲线C C / K K有K个K个有理点有无限多个K个K个有理点。Q¯\bar {\mathbb {Q}}的子域K K是大的，如果对于每一个正整数n n, K K包含一个数字域F F，且[F: Q] [F:\mathbb {Q}]能被n n整除。是否所有的大领域都是大的问题似乎已经流传了一段时间，尽管我们一直无法找到它的起源。在本文中，我们证明了存在一些不大的大场。
A subfield KK of Q¯\bar {\mathbb {Q}} is large if every smooth curve CC over KK with a KK-rational point has infinitely many KK-rational points. A subfield KK of Q¯\bar {\mathbb {Q}} is big if for every positive integer nn, KK contains a number field FF with [F:Q][F:\mathbb {Q}] divisible by nn. The question of whether all big fields are large seems to have circulated for some time, although we have been unable to find its origin. In this paper we show that there are big fields that are not large.
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