{"title":"椭圆曲线族的根号及两种应用","authors":"Jonathan Love","doi":"10.1016/j.indag.2024.04.003","DOIUrl":null,"url":null,"abstract":"<div><p>For each <span><math><mrow><mi>t</mi><mo>∈</mo><mi>Q</mi><mo>∖</mo><mrow><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span>, define an elliptic curve over <span><math><mi>Q</mi></math></span> by <span><span><span><math><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>:</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi>x</mi><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>+</mo><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mo>.</mo></mrow></math></span></span></span>Using a formula for the root number <span><math><mrow><mi>W</mi><mrow><mo>(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> as a function of <span><math><mi>t</mi></math></span> and assuming some standard conjectures about ranks of elliptic curves, we determine (up to a set of density zero) the set of isomorphism classes of elliptic curves <span><math><mrow><mi>E</mi><mo>/</mo><mi>Q</mi></mrow></math></span> whose Mordell–Weil group contains <span><math><mrow><mi>Z</mi><mo>×</mo><mi>Z</mi><mo>/</mo><mn>2</mn><mi>Z</mi><mo>×</mo><mi>Z</mi><mo>/</mo><mn>4</mn><mi>Z</mi></mrow></math></span>, and the set of rational numbers that can be written as a product of the slopes of two rational right triangles.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357724000351/pdfft?md5=2bd90ba3afb1d531934bbb073c1710e2&pid=1-s2.0-S0019357724000351-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Root numbers of a family of elliptic curves and two applications\",\"authors\":\"Jonathan Love\",\"doi\":\"10.1016/j.indag.2024.04.003\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>For each <span><math><mrow><mi>t</mi><mo>∈</mo><mi>Q</mi><mo>∖</mo><mrow><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span>, define an elliptic curve over <span><math><mi>Q</mi></math></span> by <span><span><span><math><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>:</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi>x</mi><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>+</mo><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mo>.</mo></mrow></math></span></span></span>Using a formula for the root number <span><math><mrow><mi>W</mi><mrow><mo>(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> as a function of <span><math><mi>t</mi></math></span> and assuming some standard conjectures about ranks of elliptic curves, we determine (up to a set of density zero) the set of isomorphism classes of elliptic curves <span><math><mrow><mi>E</mi><mo>/</mo><mi>Q</mi></mrow></math></span> whose Mordell–Weil group contains <span><math><mrow><mi>Z</mi><mo>×</mo><mi>Z</mi><mo>/</mo><mn>2</mn><mi>Z</mi><mo>×</mo><mi>Z</mi><mo>/</mo><mn>4</mn><mi>Z</mi></mrow></math></span>, and the set of rational numbers that can be written as a product of the slopes of two rational right triangles.</p></div>\",\"PeriodicalId\":56126,\"journal\":{\"name\":\"Indagationes Mathematicae-New Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-05-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0019357724000351/pdfft?md5=2bd90ba3afb1d531934bbb073c1710e2&pid=1-s2.0-S0019357724000351-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Indagationes Mathematicae-New Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0019357724000351\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Indagationes Mathematicae-New Series","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0019357724000351","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Root numbers of a family of elliptic curves and two applications
For each , define an elliptic curve over by Using a formula for the root number as a function of and assuming some standard conjectures about ranks of elliptic curves, we determine (up to a set of density zero) the set of isomorphism classes of elliptic curves whose Mordell–Weil group contains , and the set of rational numbers that can be written as a product of the slopes of two rational right triangles.
期刊介绍：
Indagationes Mathematicae is a peer-reviewed international journal for the Mathematical Sciences of the Royal Dutch Mathematical Society. The journal aims at the publication of original mathematical research papers of high quality and of interest to a large segment of the mathematics community. The journal also welcomes the submission of review papers of high quality.
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