As usual, let G_Q denote the absolute Galois group of Q, and let A_f denote the ring of finite adeles of Q. One can obtain a quite large and interesting representation of the product G_Q x GL₂(A_f) by p-adically completing the etale cohomology of modular curves in a suitable manner. The goal of this talk is to discuss some conjectures and open questions concerning the structure of this representation. These conjectures are closely related to recent developments in the so-called ``p-adic Langlands programme'' that is currently being developed by Breuil, Colmez, and others.