Brian Conrad, Institute for Advanced Study
"Deformation theory of Galois representations for the non-expert"
March 19, 1998
Our aim is to give some idea (to a non-number theorist) of what Mazur's
deformation theory of Galois representations is about and how its
applicability to the study of elliptic curves has been extended since
Wiles' breakthrough. The Taniyama-Shimura conjecture relates, in a
very non-trivial way, the algebraic theory of elliptic curves and the
analytic theory of modular forms. We will use Wiles' reformulation of
this conjecture in terms of deformation theory as a means of indicating
what the deformation theory can do. Extending these methods to more
elliptic curves has required some generalizations of the deformation
problems considered. We give an overview of some of this work, as well
a big remaining deformation-theoretic difficulty.