Galois representations, Moduli Spaces and Applications

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This research program belongs mainly to the area of arithmetic geometry, i.e., to the area which applies the methods of algebraic geometry to solve problems in number theory. A typical example here is the famous Fermat equation xn + yn = zn, where n > 2. Fermat asserted in 1640 that this equation has no solution in positive integers, i.e., that the sum of two n-th powers can never be an n-th power, if n > 2. This was resolved in 1995 when Wiles, using ideas and results of Frey and Ribet in arithmetic geometry, proved that this assertion is indeed true.*** In studying problems in this area, one is frequently led to the study of the arithmetic and geometry of certain moduli spaces: these are algebraic varieties (such as curves, surfaces, etc.) whose points correspond to isomorphism classes of algebraic objects (e.g. curves). For example, the points of modular curves correspond to isomorphism classes of elliptic curves with extra structure.*** A key technique in the proof of Wiles is to study what are known as Galois representations (attached to elliptic curves) and to relate them to modular forms.*** The aim of this research program is to study the arithmetic and the geometry of the moduli spaces ZN: these are surfaces whose points classify isomorphisms between certain Galois representations of elliptic curves. Of special interest here is to study the curves that lie on these moduli surfaces and to identify those that come from modular curves. In addition, it is of interest to examine the points which lie on the intersection of two such curves. Such points arise from isomorphisms of Galois representations attached to elliptic curves with complex multiplication (CM) and hence are called CM points.*** The moduli space ZN is closely connected with a certain moduli space called a Humbert surface whose points classify curves of genus 2 with an elliptic subcover of degree N. Thus, a main application of the above is to study problems involving Humbert surfaces. For example, the study of the components of the intersection of such Humbert surfaces is a problem that can be treated successfully here.*** One novel technique here is what might be called "Inverse arithmetic geometry." This consists of the systematic usage of methods and results in number theory to derive interesting results about the geometry of certain moduli spaces.*** This research has many applications, not only to number theory and to arithmetic geometry, but also to algebraic geometry (moduli spaces, Humbert schemes), to mathematical physics (Hurwitz spaces, moduli spaces), to dynamical systems (mathematical billiards) and to mirror symmetry.*** In addition, this research proposal involves highly qualified personnel (HQP) of all levels: summer undergraduate students (holding an USRA), graduate students (both M.Sc. and Ph.D. students) and post-doctoral students.