On the Theory of Higher Rank Euler, Kolyvagin and Stark Systems

Abstract

Abstract Mazur and Rubin have recently developed a theory of higher rank Kolyvagin and Stark systems over principal artinian rings and discrete valuation rings. We describe a natural extension of (a slightly modified version of) their theory to systems over more general coefficient rings. We also construct unconditionally, and for general $p$-adic representations, a canonical, and typically large, module of higher rank Euler systems and show that for $p$-adic representations satisfying standard hypotheses the image under a natural higher rank Kolyvagin-derivative-type homomorphism of each such system is a higher rank Kolyvagin system that originates from a Stark system.