Hida families for GL2 and p-adic Hodge theory

Bryden Cais

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.