Controlling Selmer groups in the higher core rank case

TL;DR

This paper extends the theory of Kolyvagin and Stark systems to arbitrary core rank, demonstrating their control over Selmer groups and establishing the freeness of the module of such systems.

Contribution

It introduces a generalized framework for Kolyvagin and Stark systems at higher core ranks, beyond the previously studied rank one case.

Findings

Kolyvagin and Stark systems control Selmer groups at arbitrary core rank.

The module of all such systems is free of rank one.

Results hold under mild hypotheses.

Abstract

We define Kolyvagin systems and Stark systems attached to $p$-adic representations in the case of arbitrary `core rank' (the core rank is a measure of the generic Selmer rank in a family of Selmer groups). Previous work dealt only with the case of core rank one, where the Kolyvagin and Stark systems are collections of cohomology classes. For general core rank, they are collections of elements of exterior powers of cohomology groups. We show under mild hypotheses that for general core rank these systems still control the size and structure of Selmer groups, and that the module of all Kolyvagin (or Stark) systems is free of rank one.