p-adic Hodge theory and values of zeta functions of modular forms

If f is a modular form, we construct an Euler attached to f from which we deduce bounds for the Selmer groups of f. An explicit reciprocity Law links this Elder system to the p-adic zeta function of f which allows us to prove a divisibility statement towards Iwasawa's main conjecture for f and to obtain lower bounds for the order of vanishing of this p-adic zeta functions. In particular, if f is associated to an elliptic curve E defined over Q, we prove that the p-adic zeta function of f has a zero at s = 1 of order at least the rank of the group of rational points on E.