The geometry of Hida families I: $$\Lambda $$-adic de Rham cohomology

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 (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 $$(\varphi ,\Gamma )$$ -modules attached to Hida’s ordinary $$\Lambda $$ -adic etale cohomology by Dee (J Algebra 235(2), 636–664, 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).