{"title":"ON THE FAMILY OF ELLIPTIC CURVES <i>X</i> + 1/<i>X</i> + <i>Y</i> + 1/<i>Y</i> + <i>t</i> = 0.","authors":"Abhishek Juyal, Dustin Moody","doi":"","DOIUrl":"","url":null,"abstract":"<p><p>We study various properties of the family of elliptic curves <i>x</i>+1/<i>x</i>+<i>y</i>+1/<i>y</i>+<i>t</i> = 0, which is isomorphic to the Weierstrass curve <dispformula> <math> <mrow><msub><mi>E</mi> <mi>t</mi></msub> <mspace></mspace> <mo>:</mo> <msup><mi>Y</mi> <mn>2</mn></msup> <mo>=</mo> <mi>X</mi> <mrow><mo>(</mo> <mrow><msup><mi>X</mi> <mn>2</mn></msup> <mo>+</mo> <mrow><mo>(</mo> <mrow> <mfrac> <mrow><msup><mi>t</mi> <mn>2</mn></msup> </mrow> <mn>4</mn></mfrac> <mo>-</mo> <mn>2</mn></mrow> <mo>)</mo></mrow> <mi>X</mi> <mo>+</mo> <mn>1</mn></mrow> <mo>)</mo></mrow> <mo>.</mo></mrow> </math> </dispformula> . This equation arises from the study of the Mahler measure of polynomials. We show that the rank of <math> <mrow><msub><mi>E</mi> <mi>t</mi></msub> <mo>(</mo> <mover><mi>Q</mi> <mo>¯</mo></mover> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo></mrow> </math> is 0 and the torsion subgroup of <math> <mrow><msub><mi>E</mi> <mi>t</mi></msub> <mo>(</mo> <mi>Q</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo></mrow> </math> is isomorphic to <math><mrow><mi>Z</mi> <mo>∕</mo> <mn>4</mn> <mi>Z</mi></mrow> </math> . Over the rational field <math><mi>Q</mi></math> we obtain infinite subfamilies of ranks (at least) one and two, and find specific instances of <i>E<sub>t</sub></i> with rank 5 and 6. We also determine all possible torsion subgroups of <math> <mrow><msub><mi>E</mi> <mi>t</mi></msub> <mo>(</mo> <mi>Q</mi> <mo>)</mo></mrow> </math> and conclude with some results regarding integral points in arithmetic progression on <i>E<sub>t</sub></i> .</p>","PeriodicalId":36228,"journal":{"name":"Integers","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8370034/pdf/nihms-1729045.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"39328304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"ELLIPTIC CURVES ARISING FROM THE TRIANGULAR NUMBERS.","authors":"Abhishek Juyal, Shiv Datt Kumar, Dustin Moody","doi":"","DOIUrl":"","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":0.0,"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":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41223260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"High rank elliptic curves with torsion ℤ/4ℤ.","authors":"Foad Khoshnam, Dustin Moody","doi":"","DOIUrl":"","url":null,"abstract":"<p><p>Working over the field ℚ(<i>t</i>), Kihara constructed an elliptic curve with torsion group ℤ/4ℤ and five independent rational points, showing the rank is at least five. Following his approach, we give a new infinite family of elliptic curves with torsion group ℤ/4ℤ and rank at least five. This matches the current record for such curves. In addition, we give specific examples of these curves with high ranks 10 and 11.</p>","PeriodicalId":36228,"journal":{"name":"Integers","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5535278/pdf/nihms875075.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"35285254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}