The Geometry of Hida Families and \Lambda-adic Hodge Theory

TL;DR

This paper develops mbda-adic de Rham and crystalline cohomologies for Hida families, establishing comparison theorems and geometric proofs of key finiteness and control results in p-adic Hodge theory.

Contribution

It introduces mbda-adic de Rham and crystalline cohomologies, providing geometric proofs of finiteness, control theorems, and new insights into the structure of Hida families.

Findings

Constructed mbda-adic de Rham and crystalline cohomologies.

Proved mbda-adic comparison isomorphisms using integral p-adic Hodge theory.

Established a geometric proof of Hida's finiteness and control theorems.

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.