{"title":"X 0 ( N ) ${X}_0(N)$ 最小正则模型的交集矩阵及其在阿拉克洛夫典范剪辑中的应用","authors":"Paolo Dolce, Pietro Mercuri","doi":"10.1112/jlms.12964","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><math>\n <semantics>\n <mrow>\n <mi>N</mi>\n <mo>></mo>\n <mn>1</mn>\n </mrow>\n <annotation>$N&gt;1$</annotation>\n </semantics></math> be an integer coprime to 6 such that <span></span><math>\n <semantics>\n <mrow>\n <mi>N</mi>\n <mo>∉</mo>\n <mo>{</mo>\n <mn>5</mn>\n <mo>,</mo>\n <mn>7</mn>\n <mo>,</mo>\n <mn>13</mn>\n <mo>}</mo>\n </mrow>\n <annotation>$N\\notin \\lbrace 5,7,13\\rbrace$</annotation>\n </semantics></math> and let <span></span><math>\n <semantics>\n <mrow>\n <mi>g</mi>\n <mo>=</mo>\n <mi>g</mi>\n <mo>(</mo>\n <mi>N</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$g=g(N)$</annotation>\n </semantics></math> be the genus of the modular curve <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>X</mi>\n <mn>0</mn>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>N</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$X_0(N)$</annotation>\n </semantics></math>. We compute the intersection matrices relative to special fibres of the minimal regular model of <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>X</mi>\n <mn>0</mn>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>N</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$X_0(N)$</annotation>\n </semantics></math>. Moreover, we prove that the self-intersection of the Arakelov canonical sheaf of <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>X</mi>\n <mn>0</mn>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>N</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$X_0(N)$</annotation>\n </semantics></math> is asymptotic to <span></span><math>\n <semantics>\n <mrow>\n <mn>3</mn>\n <mi>g</mi>\n <mi>log</mi>\n <mi>N</mi>\n </mrow>\n <annotation>$3g\\log N$</annotation>\n </semantics></math>, for <span></span><math>\n <semantics>\n <mrow>\n <mi>N</mi>\n <mo>→</mo>\n <mo>+</mo>\n <mi>∞</mi>\n </mrow>\n <annotation>$N\\rightarrow +\\infty$</annotation>\n </semantics></math>.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Intersection matrices for the minimal regular model of \\n \\n \\n \\n X\\n 0\\n \\n \\n (\\n N\\n )\\n \\n \\n ${X}_0(N)$\\n and applications to the Arakelov canonical sheaf\",\"authors\":\"Paolo Dolce, Pietro Mercuri\",\"doi\":\"10.1112/jlms.12964\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>N</mi>\\n <mo>></mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$N&gt;1$</annotation>\\n </semantics></math> be an integer coprime to 6 such that <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>N</mi>\\n <mo>∉</mo>\\n <mo>{</mo>\\n <mn>5</mn>\\n <mo>,</mo>\\n <mn>7</mn>\\n <mo>,</mo>\\n <mn>13</mn>\\n <mo>}</mo>\\n </mrow>\\n <annotation>$N\\\\notin \\\\lbrace 5,7,13\\\\rbrace$</annotation>\\n </semantics></math> and let <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>g</mi>\\n <mo>=</mo>\\n <mi>g</mi>\\n <mo>(</mo>\\n <mi>N</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$g=g(N)$</annotation>\\n </semantics></math> be the genus of the modular curve <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>X</mi>\\n <mn>0</mn>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>N</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$X_0(N)$</annotation>\\n </semantics></math>. We compute the intersection matrices relative to special fibres of the minimal regular model of <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>X</mi>\\n <mn>0</mn>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>N</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$X_0(N)$</annotation>\\n </semantics></math>. Moreover, we prove that the self-intersection of the Arakelov canonical sheaf of <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>X</mi>\\n <mn>0</mn>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>N</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$X_0(N)$</annotation>\\n </semantics></math> is asymptotic to <span></span><math>\\n <semantics>\\n <mrow>\\n <mn>3</mn>\\n <mi>g</mi>\\n <mi>log</mi>\\n <mi>N</mi>\\n </mrow>\\n <annotation>$3g\\\\log N$</annotation>\\n </semantics></math>, for <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>N</mi>\\n <mo>→</mo>\\n <mo>+</mo>\\n <mi>∞</mi>\\n </mrow>\\n <annotation>$N\\\\rightarrow +\\\\infty$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-07-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12964\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12964","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
让 N > 1 $N>1$是一个与 6 共乘的整数，使得 N ∉ { 5 , 7 , 13 }。 $Nnotin \lbrace 5,7,13\rbrace$ 并让 g = g ( N ) $g=g(N)$ 是模态曲线 X 0 ( N ) $X_0(N)$ 的属数。我们计算相对于 X 0 ( N ) $X_0(N)$ 最小正则模型的特殊纤维的交集矩阵。此外，我们还证明了在 N → + ∞ $N\rightarrow +\infty$ 时，X 0 ( N ) $X_0(N)$ 的阿拉克洛夫（Arakelov）典范 Sheaf 的自交渐近于 3 g log N $3g\log N$ 。
Intersection matrices for the minimal regular model of
X
0
(
N
)
${X}_0(N)$
and applications to the Arakelov canonical sheaf
Let be an integer coprime to 6 such that and let be the genus of the modular curve . We compute the intersection matrices relative to special fibres of the minimal regular model of . Moreover, we prove that the self-intersection of the Arakelov canonical sheaf of is asymptotic to , for .
期刊介绍：
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
