On derivatives of Kato's Euler system for elliptic curves

Abstract

In this paper, we formulate a new conjecture concerning Kato's Euler system for elliptic curves $E$ over $\mathbb{Q}$. 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 $\mathbb{Q}$. 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.