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

TL;DR

This paper proves a conjecture relating the rank of elliptic curves to special elements called Mazur-Tate elements, using divisibility properties of derivatives of Kato's Euler system.

Contribution

It establishes the Mazur-Tate refined BSD conjecture under mild assumptions by analyzing derivatives of Kato's Euler system.

Findings

Proves the Mazur-Tate conjecture relating Mordell-Weil rank and vanishing order of Mazur-Tate elements.

Demonstrates divisibility properties of derivatives of Kato's Euler system.

Provides new techniques connecting Euler systems with conjectures in BSD theory.

Abstract

Mazur and Tate proposed a conjecture which compares the Mordell-Weil rank of an elliptic curve over $\mathbb{Q}$ 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.