`-adic representations from Hurwitz spaces

The Hurwitz space approach to the regular Inverse Galois Problem was the only successful approach to Galois group realizations beyond nilpotent groups. It gave regular realizations of many series of groups. More significantly, the M(odular) T(ower) program identified obstructions to systematically finding regular realizations. Finding a way around those obstructions generalize renown results on modular curves. We use the MT program to explicitly construct towers over a number field, and analyze their cusps. Our main example looks simple at first, close to examples that attracted Riemann surface people related to Theta functions (Schottky problem). This produces families of `-adic representations. The surprising ingredient is the appearance of a Heisenberg group that controls the components that define the tower. We model our results on properties Serre used in achieving his O(pen) I(mage) T(heorem) on `-adic representations from projective systems of points on modular curves. There are two types of systems of `-adic representations: Frattini and Split. Each type generalizes aspects of modular curve systems.