$\Lambda$-adic Kolyvagin systems

TL;DR

This paper proves the existence of cyclotomic deformations of Kolyvagin systems under certain conditions, linking them to p-adic L-functions and Iwasawa theory, and explores potential extensions to other deformations.

Contribution

It establishes the existence of cyclotomic deformations of Kolyvagin systems and discusses their applications to p-adic L-functions and Iwasawa theory.

Findings

Existence of cyclotomic deformations of Kolyvagin systems under certain hypotheses.

Connections established between $oldsymbol{ m extit{ extLambda}}$-adic Kolyvagin systems and p-adic L-functions.

Implications for main conjectures and Iwasawa theory of Rubin-Stark units.

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.