  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.
