On anticyclotomic Euler and Kolyvagin systems

TL;DR

This paper develops an axiomatic framework for anticyclotomic Euler systems applicable to various Galois representations, enabling the construction of universal Kolyvagin systems and advancing understanding of Selmer groups and Iwasawa theory.

Contribution

It introduces a broad axiomatization of anticyclotomic Euler systems and demonstrates how to derive universal Kolyvagin systems from them, extending previous concrete cases.

Findings

Constructed universal Kolyvagin systems from anticyclotomic Euler systems.

Applied the framework to Selmer groups and Iwasawa main conjectures.

Reviewed concrete examples in existing literature.

Abstract

We introduce an axiomatization of the notion of ( $p$-complete) anticyclotomic Euler system for a wide class of Galois representations, including those attached to a cuspidal eigenform and to a Hida family of modular forms. Under a minimal set of assumptions, we show how to build from these data a universal Kolyvagin system for the representation and for its anticyclotomic twist. Eventually, we recover some applications to the structure of Selmer groups and Iwasawa main conjectures and we review a few concrete examples of these abstract notions that can be found in the literature.