On the theory of higher rank Euler, Kolyvagin and Stark systems, II

TL;DR

This paper establishes a canonical homomorphism linking higher rank Euler and Kolyvagin systems within the context of $p$-adic representations, advancing the understanding of their role in controlling Selmer modules.

Contribution

It proves the existence of a higher Kolyvagin derivative homomorphism, confirming a conjecture by Mazur and Rubin, and applies this to the Rubin-Stark Euler system.

Findings

Existence of a canonical homomorphism between higher rank Euler and Kolyvagin systems.

Implication that higher rank Euler systems control Selmer module structures.

Application to the conjectural Rubin-Stark Euler system.

Abstract

We prove the existence of a canonical `higher Kolyvagin derivative' homomorphism between the modules of higher rank Euler systems and higher rank Kolyvagin systems, as has been conjectured to exist by Mazur and Rubin. This homomorphism exists in the setting of $p$-adic representations that are free with respect to the action of a Gorenstein order $\mathcal{R}$ and, in particular, implies that higher rank Euler systems control the $\mathcal{R}$-module structures of Selmer modules attached to the representation. We give a first application of this theory by considering the (conjectural) Euler system of Rubin-Stark elements.