The preliminary chapter outlining the book. It emphasizes that the book presents the nilpotent theory of Riemann surface covers. This uses a combination of finite group theory and function theory extending Riemann's analysis of the Jacobi Inversion Problem, what we call the abelian theory. This nilpotent theory applies to Modular Towers, a grand generalization of modular curves that fits them naturally in all studies of Riemann surfaces and their moduli. Despite the ambitious scope, the book is firmly based in Ahlfor's graduate text on complex variables.

One example of the Modular Tower approach is the way even and odd theta nulls naturally appear as automorphic functions on the moduli spaces that are levels of particular Modular Towers attached to alternating groups and the prime p=2. This vastly generalizes Serre's Stiefel-Whitney class approach to spin cover realizations in Inverse Galois Problem. Riemann wrote three papers on half-canonical divisor classes, and his interests come alive in a modern context through this case of Modular Towers. With an elementary extension of profinite group theory using the universal p-Frattini cover of a finite group, the result is a direct relation between classical moduli of curves and most of the famous theorems of arithmetic geometry. Present research applications emphasize generalizations of Serre's Open Image Theorem for the action of the absolute Galois group of the rational numbers on projective systems of points on modular curves. We understand these as interpreting properties of Modular Tower levels using versions of Deligne's tangential base points through results of Ihara-Matsumoto-Nakamura-Wewers. As the theory develops, there appear new phenomena related to odd primes dividing a finite group. Riemann surface theorists will find a whole new world around precise Schottky-type problems for all primes. Group theorists will be surprised at the algebraic and arithmetic geometry applications of modular representation theory.