2019/10/16 by Burns, David, Kurihara, Masato, Sano, Takamichi · 1 citation
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1910.07404
In this paper we study a new conjecture concerning Kato's Euler system of zeta elements for elliptic curves E over 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 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.