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

Michael D. Fried – GJM, Volume 5, Issue 1 (2020), 38-75.

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 $\bf g$, toward studying the functions $\phi: W \to \mathbb P^1_z$ on them. Data for such pairs $(W,\phi)$ starts with a monodromy group $G$ and conjugacy classes $\bf C$ in $G$. Many applications come from putting all such covers attached to $(G,\bf C)$ 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 [58] and [63]. 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 MT s. Solutions of pieces on both ends of these connections are known in significant cases.

Categories: 2020, Issue1