On Euler systems of rank $r$ and their Kolyvagin systems

TL;DR

This paper develops a general machinery for Euler and Kolyvagin systems of rank r, providing bounds on Selmer groups that align with Bloch-Kato conjectures and applying it to p-adic L-functions.

Contribution

It introduces a unified Kolyvagin system framework for Euler systems of rank r, extending previous work and connecting to Bloch-Kato conjectures.

Findings

Provides bounds on Selmer groups using r x r determinants

Generalizes results of Kato, Rubin, Perrin-Riou

Supports Bloch-Kato conjectures through applications

Abstract

In this paper we set up a general Kolyvagin system machinery for Euler systems of rank r (in the sense of Perrin-Riou) associated to a large class of Galois representations, building on our previous work on Kolyvagin systems of Rubin-Stark units and generalizing the results of Kato, Rubin and Perrin-Riou. Our machinery produces a bound on the size of the classical Selmer group attached to a Galoys representation T (that satisfies certain technical hypotheses) in terms of a certain r \times r determinant; a bound which remarkably goes hand in hand with Bloch-Kato conjectures. At the end, we present an application based on a conjecture of Perrin-Riou on p-adic L-functions, which lends further evidence to Bloch-Kato conjectures.