Intersection matrices for the minimal regular model of X 0 ( N ) ${X}_0(N)$ and applications to the Arakelov canonical sheaf

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-07-10 DOI:10.1112/jlms.12964

Paolo Dolce, Pietro Mercuri

{"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":"Let <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 <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 <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 <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 <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 <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 <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 <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>.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12964","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}

引用次数: 0

Abstract

Let $N > 1$ be an integer coprime to 6 such that $N \notin {5, 7, 13}$ and let $g = g (N)$ be the genus of the modular curve $X_{0} (N)$ . We compute the intersection matrices relative to special fibres of the minimal regular model of $X_{0} (N)$ . Moreover, we prove that the self-intersection of the Arakelov canonical sheaf of $X_{0} (N)$ is asymptotic to $3 g \log N$ , for $N \to + \infty$ .

查看原文本刊更多论文

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

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 $3g\log N$ 。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.