My research centers on the arithmetic and geometry of modular curves, with emphasis on the following aspects:
Generalized Fermat equations. This research is motivated by the problem of resolving generalized Fermat equations using the Galois representations and modularity.
Rational points on modular varieties. My research program investigates both fundamental questions about rational points on modular varieties and their applications to diophantine problems.
Explicit representation theory. This research concerns studying explicit aspects of representation theory which are motivated by and arise from my other research interests.
Galois representations and automorphic forms. The recent advances in our knowledge about Galois representations has yielded powerful tools to study certain types of diophantine equations. This research aims to study those aspects which are relevant to diophantine applications.
Q-curves. These are natural generalizations of the notion of a rational modular elliptic curve to a general number field. They arise both from the point of view Galois representations and in applications to diophantine equations.
Drinfeld modules. 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.
Special functions. Classical special functions and special functions over finite fields appear in many disparate areas, giving explicit formulae for many important quantities (such as periods, traces of Frobenius, explicit models for representations spaces, formulae for 1/Pi, etc). This research aims to study special functions with a view to arithmetical applications.
Application motivated problems. RING-LWE and the dihedral hidden subgroup problem.
[Algebraic number theory, arithmetic geometry, representation theory; automorphic forms, diophantine problems, Drinfeld modules, elliptic curves, function fields, Galois representations, generalized Fermat equations, Q-curves, modular varieties, special functions.]