Moduli relations between l-adic representations and the regular inverse Galois problem

Abstract

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 Θ 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 the 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 (MTs). Backdrop for the OIT and RIGP uses Serre's books [Se68] and [Se92]. 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. Branch points and local monodromy2.1.2. Algebraic Geometry 2.1.3. Analytic Geometry 2.2. Part II: Braids and deforming covers 2.2.1. Dragging a cover by its branch points 2.2.2. Braids and equivalences 2.2.3. Genus formula: r = 4 2.3. Main Example 2.3.1. A lift invariant 2.3.2. A 4 , r = 4 2.3.3. The sh-incidence matrix 3. ℓ-Frattini covers and MTs 3.1. Universal ℓ-Frattini covers 3.2. Universal Frattini cover 3.3. Using ℓ Gab and 1 ℓ G 3.3.1. ℓ Gab lattice quotients 3.3.2. Nielsen classes of ℓ-Frattini lattices 3.3.3. Normal ℓ-Sylow and other cases 3.4. Test for a nonempty MT 3.4.1. The obstructed component criterion 3.4.2. Obstructed A n components 3.4.3. Finishing Ex. 2.16 3.5. Archetype of the ℓ-Frattini conjectures 4. Monodromy and ℓ-adic representations 4.1. ℓ-adic representations 4.1.1. Geometric monodromy of a MT 4.1.2. Using G → G n-lm 4.2. MTs and the OIT generalization 4.3. Using eventually ℓ-Frattini 4.4. RIGP and Faltings 4.4.1. ℓ ′ RIGP 4.4.2. GL 2 toughest point 5. The path to M(odular)T(ower)s 5.1. Historical motivations 5.1.1. Guides 5.1.2. Settings for the RIGP and acknowledgements 5.2. What Gauss told Riemann about Abel's Theorem 5.2.1. Early OIT and MT relations 5.2.2. Competition between algebra and analysis 47 5.2.3.