The Geometry of Hida Families I: $\Lambda$-adic de Rham cohomology

TL;DR

This paper develops a geometric framework for $ ext{Lambda}$-adic de Rham cohomology related to Hida families, establishing finiteness, duality, and isomorphisms with cuspforms, and sets the stage for further $p$-adic Hodge theory applications.

Contribution

It constructs the $ ext{Lambda}$-adic de Rham cohomology and proves key finiteness, control, and duality theorems using geometric methods, extending Hida's and Ohta's work.

Findings

Established $ ext{Lambda}$-adic de Rham cohomology with finiteness and duality properties.

Proved the isomorphism between $ ext{Lambda}$-adic differentials and ordinary cuspforms.

Set the foundation for crystalline and comparison theorems in the sequel.

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 $\mathbf{Q}_p$, 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 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 the sequel, we will be able to provide a "cohomological" construction of the family of $(\varphi,\Gamma)$-modules attached to Hida's ordinary $\Lambda$-adic \'etale cohomology by the work of Dee, as well as 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.