{"title":"曲线上和的牛顿多边形 I:局部到全局定理","authors":"Joe Kramer-Miller, James Upton","doi":"10.1007/s00208-024-02949-y","DOIUrl":null,"url":null,"abstract":"<p>The purpose of this article is to study Newton polygons of certain abelian <i>L</i>-functions on curves. Let <i>X</i> be a smooth affine curve over a finite field <span>\\(\\mathbb {F}_q\\)</span> and let <span>\\(\\rho :\\pi _1(X) \\rightarrow \\mathbb {C}_p^\\times \\)</span> be a finite character of order <span>\\(p^n\\)</span>. By previous work of the first author, the Newton polygon <span>\\({{\\,\\mathrm{\\text {NP}}\\,}}(\\rho )\\)</span> lies above a ‘Hodge polygon’ <span>\\({{\\,\\mathrm{\\text {HP}}\\,}}(\\rho )\\)</span> defined using ramification invariants of <span>\\(\\rho \\)</span>. In this article we study the contact between these two polygons. We prove that <span>\\({{\\,\\mathrm{\\text {NP}}\\,}}(\\rho )\\)</span> and <span>\\({{\\,\\mathrm{\\text {HP}}\\,}}(\\rho )\\)</span> share a vertex if and only if a corresponding vertex is shared between the Newton and Hodge polygons of ‘local’ <i>L</i>-functions associated to each ramified point of <span>\\(\\rho \\)</span>. As a consequence, we determine a necessary and sufficient condition for the coincidence of <span>\\({{\\,\\mathrm{\\text {NP}}\\,}}(\\rho )\\)</span> and <span>\\({{\\,\\mathrm{\\text {HP}}\\,}}(\\rho )\\)</span>.\n</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"29 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Newton polygons of sums on curves I: local-to-global theorems\",\"authors\":\"Joe Kramer-Miller, James Upton\",\"doi\":\"10.1007/s00208-024-02949-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The purpose of this article is to study Newton polygons of certain abelian <i>L</i>-functions on curves. Let <i>X</i> be a smooth affine curve over a finite field <span>\\\\(\\\\mathbb {F}_q\\\\)</span> and let <span>\\\\(\\\\rho :\\\\pi _1(X) \\\\rightarrow \\\\mathbb {C}_p^\\\\times \\\\)</span> be a finite character of order <span>\\\\(p^n\\\\)</span>. By previous work of the first author, the Newton polygon <span>\\\\({{\\\\,\\\\mathrm{\\\\text {NP}}\\\\,}}(\\\\rho )\\\\)</span> lies above a ‘Hodge polygon’ <span>\\\\({{\\\\,\\\\mathrm{\\\\text {HP}}\\\\,}}(\\\\rho )\\\\)</span> defined using ramification invariants of <span>\\\\(\\\\rho \\\\)</span>. In this article we study the contact between these two polygons. We prove that <span>\\\\({{\\\\,\\\\mathrm{\\\\text {NP}}\\\\,}}(\\\\rho )\\\\)</span> and <span>\\\\({{\\\\,\\\\mathrm{\\\\text {HP}}\\\\,}}(\\\\rho )\\\\)</span> share a vertex if and only if a corresponding vertex is shared between the Newton and Hodge polygons of ‘local’ <i>L</i>-functions associated to each ramified point of <span>\\\\(\\\\rho \\\\)</span>. As a consequence, we determine a necessary and sufficient condition for the coincidence of <span>\\\\({{\\\\,\\\\mathrm{\\\\text {NP}}\\\\,}}(\\\\rho )\\\\)</span> and <span>\\\\({{\\\\,\\\\mathrm{\\\\text {HP}}\\\\,}}(\\\\rho )\\\\)</span>.\\n</p>\",\"PeriodicalId\":18304,\"journal\":{\"name\":\"Mathematische Annalen\",\"volume\":\"29 1\",\"pages\":\"\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-07-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematische Annalen\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00208-024-02949-y\",\"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":"Mathematische Annalen","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00208-024-02949-y","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Newton polygons of sums on curves I: local-to-global theorems
The purpose of this article is to study Newton polygons of certain abelian L-functions on curves. Let X be a smooth affine curve over a finite field \(\mathbb {F}_q\) and let \(\rho :\pi _1(X) \rightarrow \mathbb {C}_p^\times \) be a finite character of order \(p^n\). By previous work of the first author, the Newton polygon \({{\,\mathrm{\text {NP}}\,}}(\rho )\) lies above a ‘Hodge polygon’ \({{\,\mathrm{\text {HP}}\,}}(\rho )\) defined using ramification invariants of \(\rho \). In this article we study the contact between these two polygons. We prove that \({{\,\mathrm{\text {NP}}\,}}(\rho )\) and \({{\,\mathrm{\text {HP}}\,}}(\rho )\) share a vertex if and only if a corresponding vertex is shared between the Newton and Hodge polygons of ‘local’ L-functions associated to each ramified point of \(\rho \). As a consequence, we determine a necessary and sufficient condition for the coincidence of \({{\,\mathrm{\text {NP}}\,}}(\rho )\) and \({{\,\mathrm{\text {HP}}\,}}(\rho )\).
期刊介绍:
Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück und Nigel Hitchin.
The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin.
Since 1868 the name Mathematische Annalen stands for a long tradition and high quality in the publication of mathematical research articles. Mathematische Annalen is designed not as a specialized journal but covers a wide spectrum of modern mathematics.