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

TL;DR

This paper explores the deep connections between l-adic Galois representations, the inverse Galois problem, and modular towers, proposing conjectures that generalize classical theorems and linking complex geometry with number theory.

Contribution

It introduces Modular Towers as a unifying framework connecting the inverse Galois problem and Serre's Open Image Theorem, extending known results beyond modular curves.

Findings

Established connections between Modular Towers and classical Galois representations

Proposed conjectures generalizing Hilbert's theorem within the Modular Towers framework

Identified cases where solutions to the inverse Galois problem are known

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 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 \phi: W -> P^1_z, on them. Data for such pairs (W,\phi) 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.