{"title":"A Lehmer-type lower bound for the canonical height on elliptic curves over function fields","authors":"Joseph H. Silverman","doi":"10.1016/j.jnt.2024.04.004","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><mi>F</mi></math></span> be the function field of a curve over an algebraically closed field with <span><math><mi>char</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>≠</mo><mn>2</mn><mo>,</mo><mn>3</mn></math></span>, and let <span><math><mi>E</mi><mo>/</mo><mi>F</mi></math></span> be a non-isotrivial elliptic curve. Then for all finite extensions <span><math><mi>K</mi><mo>/</mo><mi>F</mi></math></span> and all non-torsion points <span><math><mi>P</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>K</mi><mo>)</mo></math></span>, the <span><math><mi>F</mi></math></span>-normalized canonical height of <em>P</em> is bounded below by<span><span><span><math><msub><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>E</mi></mrow></msub><mo>(</mo><mi>P</mi><mo>)</mo><mo>≥</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>10500</mn><mo>⋅</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>F</mi></mrow></msub><msup><mrow><mo>(</mo><msub><mrow><mi>j</mi></mrow><mrow><mi>E</mi></mrow></msub><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>⋅</mo><msup><mrow><mo>[</mo><mi>K</mi><mo>:</mo><mi>F</mi><mo>]</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>.</mo></math></span></span></span></p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24001070","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
Let be the function field of a curve over an algebraically closed field with , and let be a non-isotrivial elliptic curve. Then for all finite extensions and all non-torsion points , the -normalized canonical height of P is bounded below by
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