Deformation Theory of Galois Representations
The proposed research will examine questions in algebraic number theory, mainly in the study of universal deformations of Galois representations. A classical approach to the subject of representations of Galois groups using modular forms can be found in the papers of Mazur, Ribet, Frey, Serre, and others. Professor Boston proposes a new approach in which modular forms are replaced by "life" of representations to characteristic zero. The notion of the "level" of a modular form is replaced by arithmetic data. There is a clear advantage to this approach in the study of elliptic curves over the rational numbers since its Galois representations have a lift to characteristic zero. This approach has been used successfully to prove some cases of a conjecture of Fontaine-Mazur. More applications can be expected.