On Euler systems for motives and Heegner points

TL;DR

This paper develops a broad theoretical framework for Euler systems associated with motives, proves partial results for the main conjecture, and applies these ideas to Heegner points and related conjectures in number theory.

Contribution

It introduces a general formulation of Iwasawa main conjectures for higher rank Euler systems and connects them to Heegner points and Tamagawa number conjectures.

Findings

Proved 'one half' of the Iwasawa main conjecture under mild conditions.

Formulated a conjecture on 'Darmon-type derivatives' of Euler systems.

Provided a new interpretation of the Heegner point main conjecture in terms of rank two Euler systems.

Abstract

We formulate an Iwasawa main conjecture for a higher rank Euler system for a general motive. We prove "one half" of the main conjecture under mild hypotheses. We also formulate a conjecture on "Darmon-type derivatives" of Euler systems and give an application to the Tamagawa number conjecture. Lastly, we specialize our general framework to the setting of Heegner points and give a natural interpretation of the Heegner point main conjecture in terms of rank two Euler systems.