Cette these concerne la geometrie de la correspondance de Langlands p-adique. On donne la formalisation des methodes de Emerton, qui permettrait d'etablir la conjecture de Fontaine-Mazur dans le cas general des groupes unitaires. Puis, on verifie que ce formalism est satisfait dans la cas de U(3) ou on utilise la construction de Breuil-Herzig pour la correspondence p-adique. De point de vue local, on commence l'etude de cohomologie modulo p et p-adiques de tour de Lubin-Tate pour GL_2(Q_p). En particulier, on demontre que on peut retrouver la correspondence de Langlands p-adique dans la cohomologie completee de tour de Lubin-Tate. / This thesis concerns the geometry behind the p-adic local Langlands correspondence. We give a formalism of methods of Emerton, which would permit to establish the Fontaine-Mazur conjecture in the general case for unitary groups. Then, we verify that our formalism works well in the case of U(3) where we use the construction of Breuil-Herzig as the input for the p-adic correspondence.From the local viewpoint, we start a study of the modulo p and p-adic cohomology of the Lubin-Tate tower for GL_2(Q_p). In particular, we show that we can find the local p-adic Langlands correspondence in the completed cohomology of the Lubin-Tate tower.