Introduction to moduli, l-adic representations and the Regular Version of the Inverse Galois Problem

TL;DR

This paper explores the structure of Hurwitz spaces and modular towers related to the Inverse Galois Problem, introducing new braid formulas, towers, and connections to Serre's OIT, with implications for understanding Galois representations.

Contribution

It introduces a canonical tower of Hurwitz spaces using the Universal Frattini cover and connects modular towers to Serre's OIT, expanding applications for the Inverse Galois Problem.

Findings

Braid formula for genus of Hurwitz spaces when r=4

Existence of canonical towers over Hurwitz spaces under certain conditions

Comparison of modular towers with modular curves and Serre's OIT

Abstract

Sect 1 introduces Nielsen classes attached to (G,C), where C is r conjugacy classes in a finite group G, and a braid action on them. These give reduced Hurwitz spaces, denoted H(G,C)^rd. The section concludes with a braid formula for the genus of these spaces when r = 4. If there is at least one prime l for which G is divisible by l, but has no Z/l quotient, then there is a canonical tower of reduced Hurwitz spaces over H(G,C)^rd, using the Universal Frattini cover, ~G, of G, and ~G_ab, its abelianized version. The towers are nonempty assuming C are l' classes satisfying a cohomological condition from a lift invariant. A M(odular)T(ower) is a projective sequence of components of the canonical tower. Sect 2 introduces the book [Fr18], which takes on generalizing Serre's O(pen)I(mage)T(heorem}, interpreted as the case when G is a dihedral group D_l and C is four repetitions of the involution conjugacy class. Serre's Theorem separated decomposition groups of projective sequences of points in the modular curve towers into two types: CM (complex multiplication) and GL_2. When r = 4, all MT levels are upper half-plane quotients ramified at 0 (of order 3), 1 (of order 2) and \infty (corresponding to the cusps). They are appropriate therefore to compare Serre's OIT with the cusps and decomposition group fibers. [Fr18] emphasizes new phenomena in cusps, and components, while still showing in high tower levels a valid comparison with modular curves. It also aims to show how MTs can expand the applications usual for modular curve towers, recognizing those problems directly interpret from the Inverse Galois Problem. The l-adic representations of the title come from the abelianized version of the Universal Frattini cover.