{"title":"关于椭圆曲线族X + 1/X + Y + 1/Y + t = 0。","authors":"Abhishek Juyal, Dustin Moody","doi":"","DOIUrl":null,"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":0.0000,"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":"0","resultStr":"{\"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\":null,\"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\":0.0000,\"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\":\"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}
引用次数: 0
摘要
研究了与Weierstrass曲线E t: y 2 = x (x 2 + (t 2 4 - 2) x+1)同构的椭圆曲线族x+1/x+y+ t = 0的各种性质。这个方程源于对多项式的马勒测度的研究。证明了et (Q¯(t))的秩为0,并且et (Q (t))的扭转子群同构于Z∕4z。在有理域Q上,我们得到了秩至少为1和秩至少为2的无限子族,并找到秩为5和6的Et的具体实例。我们还确定了Et (Q)的所有可能的扭转子群,并得到了关于Et上等差数列积分点的一些结果。
ON THE FAMILY OF ELLIPTIC CURVES X + 1/X + Y + 1/Y + t = 0.
We study various properties of the family of elliptic curves x+1/x+y+1/y+t = 0, which is isomorphic to the Weierstrass curve . This equation arises from the study of the Mahler measure of polynomials. We show that the rank of is 0 and the torsion subgroup of is isomorphic to . Over the rational field we obtain infinite subfamilies of ranks (at least) one and two, and find specific instances of Et with rank 5 and 6. We also determine all possible torsion subgroups of and conclude with some results regarding integral points in arithmetic progression on Et .