Ultra Kolyvagin systems and higher Fitting ideals of Iwasawa Selmer groups

Abstract

We develop the theory of equivariant, ultra Kolyvagin systems to bypass structural limitations of the Euler system machinery over infinite rings. By utilizing collections of classes living in the exterior powers of patched Selmer groups -- constructed from ultraproducts of classical Selmer groups -- we compute the structure of an Iwasawa Selmer group up to pseudo-isomorphism of Iwasawa modules and prove the absence of finite submodules. We apply this theoretical framework to the fine Selmer group of an elliptic curve and the Bloch-Kato Selmer group of the Rankin-Selberg convolution of modular forms.