{"title":"由三角数产生的椭圆曲线。","authors":"Abhishek Juyal, Shiv Datt Kumar, Dustin Moody","doi":"","DOIUrl":null,"url":null,"abstract":"<p><p>We study the Legendre family of elliptic curves <i>E<sub>t</sub></i> : <i>y</i> <sup>2</sup> = <i>x</i>(<i>x</i> - 1)(<i>x</i> - Δ <sub><i>t</i></sub> ), parametrized by triangular numbers Δ <sub><i>t</i></sub> = <i>t</i>(<i>t</i> + 1)/2. We prove that the rank of <i>E<sub>t</sub></i> over the function field <math> <mrow><mover><mi>Q</mi> <mo>‒</mo></mover> <mo>(</mo> <mi>t</mi> <mo>)</mo></mrow> </math> is 1, while the rank is 0 over <math><mrow><mi>Q</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo></mrow> </math> . We also produce some infinite subfamilies whose Mordell-Weil rank is positive, and find high rank curves from within these families.</p>","PeriodicalId":36228,"journal":{"name":"Integers","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6604644/pdf/nihms-1526050.pdf","citationCount":"0","resultStr":"{\"title\":\"ELLIPTIC CURVES ARISING FROM THE TRIANGULAR NUMBERS.\",\"authors\":\"Abhishek Juyal, Shiv Datt Kumar, Dustin Moody\",\"doi\":\"\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>We study the Legendre family of elliptic curves <i>E<sub>t</sub></i> : <i>y</i> <sup>2</sup> = <i>x</i>(<i>x</i> - 1)(<i>x</i> - Δ <sub><i>t</i></sub> ), parametrized by triangular numbers Δ <sub><i>t</i></sub> = <i>t</i>(<i>t</i> + 1)/2. We prove that the rank of <i>E<sub>t</sub></i> over the function field <math> <mrow><mover><mi>Q</mi> <mo>‒</mo></mover> <mo>(</mo> <mi>t</mi> <mo>)</mo></mrow> </math> is 1, while the rank is 0 over <math><mrow><mi>Q</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo></mrow> </math> . We also produce some infinite subfamilies whose Mordell-Weil rank is positive, and find high rank curves from within these families.</p>\",\"PeriodicalId\":36228,\"journal\":{\"name\":\"Integers\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6604644/pdf/nihms-1526050.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Integers\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"\",\"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":"Integers","FirstCategoryId":"1085","ListUrlMain":"","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}
ELLIPTIC CURVES ARISING FROM THE TRIANGULAR NUMBERS.
We study the Legendre family of elliptic curves Et : y2 = x(x - 1)(x - Δ t ), parametrized by triangular numbers Δ t = t(t + 1)/2. We prove that the rank of Et over the function field is 1, while the rank is 0 over . We also produce some infinite subfamilies whose Mordell-Weil rank is positive, and find high rank curves from within these families.