Ohta described the ordinary part of the \'etale cohomology of towers of modular curves in terms of Hida families. Ohta's approach crucially depended on the one-dimensional nature of modular curves. In this talk, I will present joint work with Chris Skinner which generalizes Ohta's results to other Shimura varieties; I will focus on the case of Hilbert modular varieties. This generalization is expected to have many applications for the construction of Euler systems. Our approach is based on an idea of Faltings, clarified for us by B. Bhatt under the framework of integral p-adic Hodge theory, as developed by Bhatt--Morrow--Scholze, Bhatt--Scholze and Bhatt--Lurie.