  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.
