{"title":"椭圆𝑗不变式的模态多项式系数的明确界限","authors":"Florian Breuer, Fabien Pazuki","doi":"10.1090/bproc/179","DOIUrl":null,"url":null,"abstract":"<p>We obtain an explicit upper bound on the size of the coefficients of the elliptic modular polynomials <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Phi Subscript upper N\">\n <mml:semantics>\n <mml:msub>\n <mml:mi mathvariant=\"normal\">Φ</mml:mi>\n <mml:mi>N</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\Phi _N</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> for any <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N greater-than-or-equal-to 1\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>N</mml:mi>\n <mml:mo>≥</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">N\\geq 1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. These polynomials vanish at pairs of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"j\">\n <mml:semantics>\n <mml:mi>j</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">j</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-invariants of elliptic curves linked by cyclic isogenies of degree <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N\">\n <mml:semantics>\n <mml:mi>N</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">N</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. The main term in the bound is asymptotically optimal as <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N\">\n <mml:semantics>\n <mml:mi>N</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">N</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> tends to infinity.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"42 5","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Explicit bounds on the coefficients of modular polynomials for the elliptic 𝑗-invariant\",\"authors\":\"Florian Breuer, Fabien Pazuki\",\"doi\":\"10.1090/bproc/179\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We obtain an explicit upper bound on the size of the coefficients of the elliptic modular polynomials <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper Phi Subscript upper N\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mi mathvariant=\\\"normal\\\">Φ</mml:mi>\\n <mml:mi>N</mml:mi>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\Phi _N</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> for any <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper N greater-than-or-equal-to 1\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>N</mml:mi>\\n <mml:mo>≥</mml:mo>\\n <mml:mn>1</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">N\\\\geq 1</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. These polynomials vanish at pairs of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"j\\\">\\n <mml:semantics>\\n <mml:mi>j</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">j</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-invariants of elliptic curves linked by cyclic isogenies of degree <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper N\\\">\\n <mml:semantics>\\n <mml:mi>N</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">N</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. The main term in the bound is asymptotically optimal as <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper N\\\">\\n <mml:semantics>\\n <mml:mi>N</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">N</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> tends to infinity.</p>\",\"PeriodicalId\":106316,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society, Series B\",\"volume\":\"42 5\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-07-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society, Series B\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/bproc/179\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/bproc/179","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
对于任意 N ≥ 1 N\geq 1，我们得到了椭圆模态多项式 Φ N \Phi _N 的系数大小的明确上限。这些多项式在椭圆曲线的 j j - 变项对上消失，这些变项通过 N N 度的循环同源关系相连。当 N N 趋于无穷大时，约束中的主项是渐近最优的。
Explicit bounds on the coefficients of modular polynomials for the elliptic 𝑗-invariant
We obtain an explicit upper bound on the size of the coefficients of the elliptic modular polynomials ΦN\Phi _N for any N≥1N\geq 1. These polynomials vanish at pairs of jj-invariants of elliptic curves linked by cyclic isogenies of degree NN. The main term in the bound is asymptotically optimal as NN tends to infinity.
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