曲线上和的牛顿多边形 I:局部到全局定理

IF 1.3 2区 数学 Q1 MATHEMATICS Mathematische Annalen Pub Date : 2024-07-29 DOI:10.1007/s00208-024-02949-y
Joe Kramer-Miller, James Upton
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引用次数: 0

摘要

本文的目的是研究曲线上某些无边 L 函数的牛顿多边形。让 X 是一条有限域上的光滑仿射曲线,让 \(\rho :\pi _1(X) \rightarrow \mathbb {C}_p^\times \) 是一个阶为 \(p^n\) 的有限特征。根据第一作者之前的研究,牛顿多边形({{\,\mathrm{text {NP}}\,}}(\rho )\) 位于使用 \(\rho \) 的斜切不变式定义的 "霍奇多边形"({{\,\mathrm{text {HP}}\,}}(\rho )\) 的上方。本文将研究这两个多边形之间的接触。我们证明当且仅当与\(\rho \)的每个斜切点相关联的 "局部 "L函数的牛顿多边形和霍奇多边形共享一个顶点时,\({{\,\mathrm{text {NP}\,}}(\rho )\) 和\({{\,\mathrm{text {HP}\,}}(\rho )\) 共享一个顶点。)因此,我们确定了 \({{\,\mathrm{text {NP}\,}}(\rho )\) 和 \({{\,\mathrm{text {HP}\,}}(\rho )\) 重合的必要条件和充分条件。)
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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 \({{\,\mathrm{\text {HP}}\,}}(\rho )\).

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来源期刊
Mathematische Annalen
Mathematische Annalen 数学-数学
CiteScore
2.90
自引率
7.10%
发文量
181
审稿时长
4-8 weeks
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.
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