Bryden Cais (McGill University)
Hida Families for GL_2 and p-adic Hodge theory
In the 1980's, Hida constructed p-adic analytic families of
ordinary Galois representations via a detailed study of Hecke algebras and
group cohomology. Shortly after this, Mazur and Wiles gave a geometric
interpretation of the associated families of Galois representations by
realizing them in the etale cohomology groups of towers of modular curves.
In accordance with the philosophy of p-adic Hodge theory, one expects
that there should be a corresponding geometric construction of p-adic
families of ordinary modular forms via de Rham cohomology. In this
talk, we will explain such a construction; as a consequence, we obtain
a new and purely geometric approach to Hida theory. Using recent
progress in integral p-adic Hodge theory, we will elucidate how our
construction can be used to recover that of Mazur-Wiles.