A generalization of Darmon's conjecture for Euler systems for general p-adic representations

TL;DR

This paper generalizes Darmon's conjecture to a broader class of p-adic representations, introducing algebraic Kolyvagin systems and proving a non-explicit version of the conjecture.

Contribution

It formulates and proves a non-explicit generalization of Darmon's conjecture for Euler systems associated with general p-adic representations, introducing algebraic Kolyvagin systems.

Findings

Established a non-explicit version of Darmon's conjecture for general p-adic representations.

Introduced and developed the theory of algebraic Kolyvagin systems.

Proved the formulated conjecture using these new algebraic tools.

Abstract

Darmon's conjecture on a relation between cyclotomic units over real quadratic fields and certain algebraic regulators was recently solved by Mazur and Rubin by using their theory of Kolyvagin systems. In this paper, we formulate a "non-explicit" version of Darmon's conjecture for Euler systems defined for general $p$-adic representations, and prove it. In the process of the proof, we introduce a notion of "algebraic Kolyvagin systems", and develop their properties.