Ultra-Kolyvagin systems and non-ordinary Selmer groups

TL;DR

This paper introduces a new method for bounding Selmer groups of Galois representations in non-ordinary settings using Euler systems and ultrafilter techniques, leading to progress on the Iwasawa main conjecture.

Contribution

It develops a novel framework combining Euler systems, Pottharst's Selmer groups, and ultrafilter methods to address non-ordinary cases in Iwasawa theory.

Findings

Proves new cases of the cyclotomic Iwasawa main conjecture for non-ordinary Rankin--Selberg convolutions.

Establishes a machine for bounding Selmer groups in non-ordinary settings.

Introduces ultrafilter-based interpretation of Kolyvagin derivative classes.

Abstract

We develop a machine for bounding Selmer groups of Galois representations via Euler systems in "non-ordinary" settings, using Pottharst's definition of Selmer groups via Robba-ring $(\varphi, \Gamma)$-modules. Our approach relies on Sweeting's interpretation of Kolyvagin derivative classes via non-principal ultrafilters. We apply these results to prove new cases of the cyclotomic Iwasawa main conjecture for non-ordinary Rankin--Selberg convolutions.