Construction of some families of 2-dimensional crystalline representations

We construct explicitly some analytic families of étale (φ,γ)-modules, which give rise to analytic families of 2-dimensional crystalline representations, As an application of our constructions, we verify some conjectures of Breuil on the reduction modulo p of those representations, and extend some results (of Deligne, Edixhoven, Fontaine and Serre) on the representations arising from modular forms.