In this series of two articles we investigate a new conjecture concerning Kato’s Euler system of zeta elements for elliptic curves E over Q. This conjecture predicts a precise congruence relation between a ‘Darmon-type’ derivative of the zeta element of E over any given abelian field and the value at the critical point of an appropriate higher derivative of the L-function of E over Q. We show 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 and also simultaneously refines (in arbitrary rank) the conjecture of Mazur and Tate concerning congruences for modular elements. In particular, by using this approach we shall obtain a proof, under certain mild and natural hypotheses, that the Mazur-Tate Conjecture is valid in analytic rank one. Under more general hypotheses we shall also prove the ‘order of vanishing’ part of the new conjecture in arbitrary rank. In addition, an Iwasawa-theoretic analysis of the approach leads, via a higher rank generalization of Rubin’s formula concerning derivatives of p-adic L-functions, to a proof (without assuming E has good reduction at p) that the main conjecture implies the validity, up to a p-adic unit, of the p-adic Birch and Swinnerton-Dyer Formula and to an understanding of the precise conditions, in arbitrary rank and with arbitrary reduction type, under which a suitable main conjecture implies the classical Birch and Swinnerton-Dyer Formula.
|Publication status||Published - 2019 Oct 16|
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