$\Lambda$-adic Kolyvagin systems

Abstract

In this paper, we study the deformations of Kolyvagin systems that are known to exist in a wide variety of cases, by the work of B. Howard, B. Mazur, and K. Rubin for the residual Galois representations, along the cyclotomic Iwasawa algebra. We prove, under certain technical hypotheses, that a cyclotomic deformation of a Kolyvagin system exists. We also briefly discuss how our techniques could be extended to prove that one could deform Kolyvagin systems for other deformations as well. We discuss several applications of this result, particularly relation of these $\Lambda$-adic Kolyvagin systems to p-adic L-functions (in view of the conjectures of Perrin-Riou on p-adic L-functions) and applications to main conjectures; also applications to the study of Iwasawa theory of Rubin-Stark units.