The goal of these lectures is to present a proof
towards some results predicted by the Bloch-Kato conjecture giving a link between the order of vanishing of the L-function of a motive and the rank of the corresponding Bloch-Kato-Selmer group.

The first lecture will be a review of the general conjecture and
its link to the Birch and
Swinnerton-Dyer conjecture. The next lectures will be devoted
to a review of the tools that are needed for the proof, namely the theory of Eisenstein series, Eigenvarieties and some p-adic Hodge theory. Some very simple case of the strategy will be studied before the last lectures that will be devoted to the proof of the following result:
Let M be an automorphic polarized motive over an imaginary quadratic field such that its L-function vanishes at the center of the functional equation, then the Bloch-Kato-Selmer group of M is of rank
at least one.