On derivatives of Kato's Euler system for elliptic curves

In this paper we study a new conjecture concerning Kato's Euler system of zeta elements for elliptic curves $E$ over $\mathbb{Q}$. This conjecture, which we refer to as the `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 $\mathbb{Q}$. We prove that the conjecture specializes in the relevant case of analytic rank one to recover Perrin-Riou's conjecture on the logarithm of Kato's zeta element. Under mild hypotheses we also prove that the `order of vanishing' part of the conjecture is valid in arbitrary rank. An Iwasawa-theoretic analysis of our approach leads to the formulation and proof of a natural higher rank generalization of Rubin's formula concerning derivatives of $p$-adic $L$-functions. In addition, we establish a concrete and apparently new 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 forthcoming paper we will show that the Generalized Perrin-Riou Conjecture implies (in arbitrary rank) the conjecture of Mazur and Tate concerning congruences for modular elements and, by using this approach, we are able to give a proof, under certain mild and natural hypotheses, that the Mazur-Tate Conjecture is valid in analytic rank one.