Ultra-Kolyvagin systems and non-ordinary Selmer groups
David Loeffler, Sarah Livia Zerbes
TL;DR
The article develops a framework to bound Selmer groups in non-ordinary settings by combining Euler systems with Pottharst’s $(\varphi,\Gamma)$-module Selmer groups and Sweeting’s ultrafilter patching to create ultra-Kolyvagin systems. It builds a robust local theory of cohomology, regulator maps, and integral structures in the Robba-ring setting, proving that under suitable hypotheses one can derive divisibilities between characteristic ideals in Iwasawa theory for non-ordinary Rankin–Selberg contexts. The main contributions include establishing topological and integral control of Selmer groups via regulator maps, formulating a precise descent to Iwasawa towers, and proving a divisibility statement for the Rankin–Selberg case, with broad potential generalisations to Asai, $\mathrm{GSp}_4$, and related automorphic settings. The results significantly extend the reach of Euler-system techniques to non-ordinary primes, enabling new cases of the cyclotomic Iwasawa main conjecture and tighter divisibility statements in analytic Iwasawa theory.
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 $(\varphi, Γ)$-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.