{"title":"The average number of integral points on the congruent number curves","authors":"Stephanie Chan","doi":"10.1016/j.aim.2024.109946","DOIUrl":null,"url":null,"abstract":"<div><p>We show that the total number of non-torsion integral points on the elliptic curves <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>:</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>x</mi></math></span>, where <em>D</em> ranges over positive squarefree integers less than <em>N</em>, is <span><math><mi>O</mi><mo>(</mo><mi>N</mi><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>N</mi><mo>)</mo></mrow><mrow><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>+</mo><mi>ϵ</mi></mrow></msup><mo>)</mo></math></span>. The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of Heath-Brown's method on estimating the average size of the 2-Selmer groups of the curves in this family.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"457 ","pages":"Article 109946"},"PeriodicalIF":1.5000,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0001870824004614/pdfft?md5=e61e01dc3d1a09b4e1bf01af1246df6b&pid=1-s2.0-S0001870824004614-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0001870824004614","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2024/9/11 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
We show that the total number of non-torsion integral points on the elliptic curves , where D ranges over positive squarefree integers less than N, is . The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of Heath-Brown's method on estimating the average size of the 2-Selmer groups of the curves in this family.
期刊介绍：
Emphasizing contributions that represent significant advances in all areas of pure mathematics, Advances in Mathematics provides research mathematicians with an effective medium for communicating important recent developments in their areas of specialization to colleagues and to scientists in related disciplines.
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