Rational points on modular curves and diophantine problems. Recent advances in the study of the generalized Fermat equation and variants have revealed a close connection with the determination of rational points on modular curves. My research program investigates both fundamental questions about rational points on modular curves and their applications to diophantine problems. My recent work in this direction has revealed several new classes of diophantine equations which can be successfully analyzed using the modular method. Further work which elucidates the applicability of this method and its fundamental nature is still required.
Applications of representation theory. From the point of view of known effective methods for the determination of rational points on modular curves, obstacles to the complete determination of rational points appear to be intrinsically group-theoretical in nature. This has required the study of the representation theory of algebraic groups over local rings and has spawned representation theoretic questions of independent interest.
Galois representations. The recent advances in our knowledge about Galois representations has yielded powerful tools to study certain types of diophantine equations. However, we are still far from a very complete understanding about the nature of modular representations in more general settings. A new direction in my research program has been to explore some of these situations, in particular the study of mod p^m representations.
Q-curves. These are natural generalizations of the notion of a rational modular elliptic curve over a general number field. They arise both from the point of view Galois representations and in applications to diophantine equations.
Drinfeld modular curves and forms. A common theme which has developed over the past decades has been the analogy between number fields and function fields. Typically, the function field case is more readily resolved and provides insight into the number field case. A recent research interest has been the arithmetic and geometric study of function field analogs of modular curves and forms.
Explicit computational methods. There are several computational aspects to my research, including the explicit computation of equations for modular curves and the computation of modular forms. The development of these explicit methods is often essential in determining key theoretical quantities used in the study of modular curves and applications to diophantine problems.
Keywords: [Algebraic number theory, arithmetic geometry, representation theory, modular varieties, automorphic forms, diophantine problems, Galois representations, elliptic curves, Q-curves, function fields.]