Starting with the work of Katz and Serre and developed by Hida one has seen that modular forms naturally live in families and that this point of view has vast arithmetic applications. In particular, these ideas have been applied by Wiles and Taylor toward a proof Fermat's Last Theorem and by Buzzard and Taylor toward new cases of Artin's conjecture. They instigated Mazur's theory of deformation spaces of residual representations which was pivotal in the aforementioned work of Taylor-Wiles. We have shown with Mazur that there is a natural curve called the eigencurve of finite slope forms whose properties have already shed light on some of the previous topics. This curve maps into the above deformation space but its image is still very mysterious. With William Stein we have shown that some points in the deformation space associated to modular forms of infinite slope are in the topological closure of the image of the eigencurve and some are not. Our research is directed at a better understanding of the image of the eigencurve.
This is a proposal in the area of mathematics called arithmetic algebraic geometry; this is where the techniques and questions of number theory merge with the techniques and questions of algebraic geometry. The main focus of this research is to better understand a particular special kind of algebraic geometric curve, called the "eigencurve," which is intimately and directly connected with a continuously varying collection of deep and important number theoretic data. Understanding the link between the eigencurve and its corresponding number theoretic data will advance the field of number theory, which in turn provides the underpinning for most of modern cryptography and digital security.
