{"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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"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":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0019357724000351","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
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.
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