John Cullinan , Shanna Dobson , Linda Frey , Asimina S. Hamakiotes , Roberto Hernandez , Nathan Kaplan , Jorge Mello , Gabrielle Scullard
{"title":"The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, II","authors":"John Cullinan , Shanna Dobson , Linda Frey , Asimina S. Hamakiotes , Roberto Hernandez , Nathan Kaplan , Jorge Mello , Gabrielle Scullard","doi":"10.1016/j.jnt.2024.07.013","DOIUrl":null,"url":null,"abstract":"<div><p>Let <em>E</em> and <span><math><msup><mrow><mi>E</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> be 2-isogenous elliptic curves over <strong>Q</strong>. Following <span><span>[6]</span></span>, we call a prime of good reduction <em>p anomalous</em> if <span><math><mi>E</mi><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo><mo>≃</mo><msup><mrow><mi>E</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> but <span><math><mi>E</mi><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>)</mo><mo>≄</mo><msup><mrow><mi>E</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>)</mo></math></span>. Our main result is an explicit formula for the proportion of anomalous primes for any such pair of elliptic curves. We consider both the CM case and the non-CM case.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001720/pdfft?md5=f8f53d9d54ebb568a03018d889d8244b&pid=1-s2.0-S0022314X24001720-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24001720","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 E and be 2-isogenous elliptic curves over Q. Following [6], we call a prime of good reduction p anomalous if but . Our main result is an explicit formula for the proportion of anomalous primes for any such pair of elliptic curves. We consider both the CM case and the non-CM case.
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