论文标题
L-ADIC表示与常规逆Galois问题之间的模量关系
Moduli relations between l-adic representations and the regular inverse Galois problem
论文作者
论文摘要
有两个著名的亚伯定理。最著名的是他对一维紧凑型复杂圆环的Abelian（分析）功能的描述。另一个将那些具有质量同学的复杂摩根集中到一个空间中。里曼（Riemann）对第一个的概括具有他著名的theta功能。他最深入的工作旨在扩展亚伯的第二个定理；他在实现这一目标之前就死了。该扩展通常是在复杂的更高尺寸托里（Torii）上所示的。但是，对于Riemann而言，这是针对紧凑型Riemann表面W属的Jacobians的空间，用于研究函数ϕ：W-> p^1_z，上面。此类对的数据（W，ϕ）始于G中的一个单型组，G和Chugacy类C。许多应用来自将所有此类覆盖物放在（G，c）中的所有这些覆盖物中，自然 - Hurwitz-家庭。我们连接两个这样的应用程序：常规的逆galois问题（RIGP）和SERRE的开放图像定理（OIT）。我们称连接设备模块化塔（MT S）。 OIT和RIGP的背景使用Serre的书籍Abelian L-Adic表示和椭圆形曲线（1968）和Galois Theory中的主题（1992）。 Serre的OIT示例是MT水平识别为模块化曲线的情况。我们以一个不是模块化曲线的示例，我们解释了猜想的MT属性（概括了希尔伯特的定理），该特性将为所有MT提供一个OIT。在重要情况下，这些连接两端的零件解决方案是已知的。
There are two famous Abel Theorems. Most well-known is his description of abelian (analytic) functions on a one dimensional compact complex torus. The other collects together those complex tori, with their prime degree isogenies, into one space. Riemann's generalization of the first features his famous theta functions. His deepest work aimed at extending Abel's second theorem; he died before he fulfilled this. That extension is often pictured on complex higher dimension torii. For Riemann, though, it was to spaces of Jacobians of compact Riemann surfaces, W, of genus g, toward studying functions ϕ: W -> P^1_z, on them. Data for such pairs (W,ϕ) starts with a monodromy group, G, and conjugacy classes C in G. Many applications come from putting all such covers attached to (G,C) in natural -- Hurwitz -- families. We connect two such applications: The Regular Inverse Galois Problem (RIGP) and Serre's Open Image Theorem (OIT). We call the connecting device Modular Towers (MT s). Backdrop for the OIT and RIGP uses Serre's books Abelian l-adic representations and elliptic curves (1968) and Topics in Galois theory (1992). Serre's OIT example is the case where MT levels identify as modular curves. With an example that isn't modular curves, we explain conjectured MT properties -- generalizing a Theorem of Hilbert's -- that would conclude an OIT for all MTs. Solutions of pieces on both ends of these connections are known in significant cases.
