Modular curves are moduli spaces for structures on elliptic curves. Modular functions come from the homogeneous space structure-through SL(2,R) action- on an affine subset of the modular curve. This award provides funds for an investigation that applies modular curve thinking to moduli spaces that have no homogeneous space structure. The key definition is a Modular Tower attached to the data coming from any finite group G and a prime p dividing the order of G. The investigation will cover applications to the following topics. (a) Invariants attached to theta functions of specific types of curves. (b) Curves over finite fields with special relations among their zeta function zeros. (c) Structured conclusions on the inverse Galois problem. (d) A program for analyzing properties of the Grothendieck- Teichmuller group using projective systems of real points on a modular tower. The objective is to show that modular towers produce solutions to classical problems not previously benefiting from analogs of modular functions. This award also supports an investigation by postdoctoral student Y. Kopeliovich. Kopeliovich will continue using theta functions with characteristic for giving detailed degree bounds on maps from modular curves to the sphere. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems. This particular proposal has applications to cryptography, through the classification and production of polynomial functions, called exceptional, specifically suited for encoding data.
