Kato's Euler system and the Mazur-Tate refined conjecture of BSD type

Abstract

Mazur and Tate proposed a conjecture which compares the Mordell-Weil rank of an elliptic curve over Q with the order of vanishing of Mazur-Tate elements, which are analogues of Stickelberger elements. Under some relatively mild assumptions, we prove this conjecture. Our strategy of the proof is to study divisibility of certain derivatives of Kato's Euler system. CONTENTS 1. Introduction 1 2. Mazur-Tate elements 4 3. Darmon-Kolyvagin derivatives and Euler systems 6 4. Divisibility of Euler systems for elliptic curves 15 5. Local study of Mazur-Tate elements 28 6. Proof of the main result 36 7. Trivial zeros 39