Λ-adic Kolyvagin Systems

In this paper, we study Kolyvagin systems, as defined by Mazur and Rubin, over the cyclotomic $\mathbb Z_p$-tower for a Gal${\mathrm Gal}(\bar {\mathbb Q}/\mathbb Q)$-representation T. We prove, under certain hypotheses, that the module of Λ-adic Kolyvagin systems for the cyclotomic deformation T⊗Λ is free of rank 1 over the cyclotomic Iwasawa algebra. We link our result with a web of conjectures due to Perrin-Riou and Rubin; and we relate the Λ-adic Kolyvagin systems we prove to exist to (conjectural) p-adic L-functions. We also study the Iwasawa theory of Rubin–Stark elements via the perspective offered by our main theorem, and outline a strategy to deduce the main conjectures of Iwasawa theory for totally real number fields assuming the Rubin–Stark conjecture.