The geometry of Hida families I: Λ-adic de Rham cohomology

Abstract

We construct the Lambda-adic de Rham analogue of Hida's ordinary Lambda-adic etale cohomology and of Ohta's Lambda-adic Hodge cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of Qp, we give a purely geometric proof of the expected finiteness, control, and Lambda-adic duality theorems. Following Ohta, we then prove that our Lambda-adic module of differentials is canonically isomorphic to the space of ordinary Lambda-adic cuspforms. In the sequel (Cais, Compos Math, to appear) to this paper, we construct the crystalline counterpart to Hida's ordinary Lambda-adic etale cohomology, and employ integral p-adic Hodge theory to prove Lambda-adic comparison isomorphisms between all of these cohomologies. As applications of our work in this paper and (Cais, Compos Math, to appear), we will be able to provide a " cohomological" construction of the family of (phi, Gamma)-modules attached to Hida's ordinary Lambda-adic etale cohomology by Dee (J Algebra 235(2), 636664, 2001), as well as a new and purely geometric proof of Hida's finiteness and control theorems. We are also able to prove refinements of the main theorems in Mazur and Wiles (Compos Math 59(2): 231-264, 1986) and Ohta (J Reine Angew Math 463: 49-98, 1995).