de Rham cohomology and the geometry of Hida families

Bryden Cais

In the 1980's, Hida constructed certain 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 work of Ohta, we will elucidate how our construction relates to that of Mazur-Wiles via standard comparison isomorphisms in p-adic Hodge theory