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

Burns, David; Sakamoto, Ryotaro; Sano, Takamichi

doi:10.2140/ant.2026.20.1677

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

David Burns, Ryotaro Sakamoto and Takamichi Sano

Vol. 20 (2026), No. 9, 1677–1726

DOI: 10.2140/ant.2026.20.1677

Abstract

We prove the existence of a canonical “higher Kolyvagin derivative” map between Euler and Kolyvagin systems in any given rank, as conjectured by Mazur and Rubin. This map is constructed in the setting of $p$ -adic representations that are free with respect to the action of a Gorenstein order $ℛ$ and implies higher rank Euler systems control the $ℛ$ -module structures of Selmer modules attached to the representation.

Keywords

Euler systems, Kolyvagin systems, Stark systems, higher rank, Kolyvagin derivative, Selmer groups

Mathematical Subject Classification

Primary: 11F80, 11G40

Secondary: 11R23, 11R34

Milestones

Received: 7 July 2024

Revised: 4 May 2025

Accepted: 5 August 2025

Published: 19 October 2026

Authors

David Burns
Department of Mathematics King’s College London London United Kingdom

Ryotaro Sakamoto
Department of Mathematics University of Tsukuba Tsukuba Japan

Takamichi Sano
Department of Mathematics Osaka Metropolitan University Sugimoto Osaka Japan

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Vol. 20, No. 9, 2026