On derivatives of Kato's Euler system for elliptic curves

In this paper, we formulate a new conjecture concerning Kato's Euler system for elliptic curves E over Q. This `Generalized Perrin-Riou Conjecture' predicts a precise congruence relation between a Darmon-type derivative of the zeta element of E over an arbitrary real abelian field and the critical value of an appropriate higher derivative of the L-function of E over Q. We prove the conjecture specializes in the relevant case of analytic rank one to recover Perrin-Riou's conjecture on the logarithms of zeta elements, and also that, under mild technical hypotheses, the `order of vanishing' part of the conjecture is unconditionally valid in arbitrary rank. This approach also allows us to prove a natural higher-rank generalization of Rubin's formula concerning derivatives of p-adic L-functions and to establish an explicit connection between the p-part of the classical Birch and Swinnerton-Dyer formula and the Iwasawa main conjecture in arbitrary rank and for arbitrary reduction at p. In a companion article we prove that the approach developed here also provides a new interpretation of the Mazur -Tate conjecture that leads to the first (unconditional) theoretical evidence in support of this conjecture for curves of strictly positive rank.