The Geometry of Hida Families and Λ-adic Hodge Theory

Abstract: We construct \Lambda-adic de Rham and crystalline analogues of Hida's ordinary \Lambda-adic etale cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of \Q_p, we prove appropriate finiteness and control theorems in each case. We then employ integral p-adic Hodge theory to prove \Lambda-adic comparison isomorphisms between our cohomologies and Hida's etale cohomology. As applications of our work, we provide a "cohomological" construction of the family of (\phi,\Gamma)-modules attached to Hida's ordinary \Lambda-adic etale cohomology by Dee, and we give a new and purely geometric proof of Hida's finitenes and control theorems. We are also able to prove refinements of theorems of Mazur-Wiles and of Ohta; in particular, we prove that there is a canonical isomorphism between the module of ordinary \Lambda-adic cuspforms and the part of the crystalline cohomology of the Igusa tower on which Frobenius acts invertibly.