  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.
