Ultra-Kolyvagin systems and non-ordinary Selmer groups

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 $(\phi ,\mathrm{\Gamma })$-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.