John Cullinan , Shanna Dobson , Linda Frey , Asimina S. Hamakiotes , Roberto Hernandez , Nathan Kaplan , Jorge Mello , Gabrielle Scullard
{"title":"有限域扩展中同源椭圆曲线非同构群结构的概率，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":"{\"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}","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}
引用次数: 0
摘要
继[6]之后，如果 E(Fp)≃E′(Fp)，但 E(Fp2)≄E′(Fp2)，我们就称好还原的素数 p 为反常素数。我们的主要结果是为任何这样一对椭圆曲线的反常素数比例提供了一个明确的公式。我们同时考虑了 CM 情况和非 CM 情况。
The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, II
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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