Euler systems with local conditions

TL;DR

This paper surveys known Euler systems, discusses conjectures about their expected forms, and proposes a broader conjecture accommodating recent unexpected constructions, highlighting multiple Euler systems of varying ranks for Galois representations.

Contribution

It introduces a new, more general conjecture predicting multiple Euler systems of different ranks for a single Galois representation, expanding current understanding.

Findings

Recent constructions of Euler systems challenge existing conjectures.

A new conjecture predicts multiple Euler systems of various ranks.

The paper suggests relationships between these multiple Euler systems.

Abstract

Euler systems are certain compatible families of cohomology classes, which play a key role in studying the arithmetic of Galois representations. We briefly survey the known Euler systems, and recall a standard conjecture of Perrin-Riou predicting what kind of Euler system one should expect for a general Galois representation. Surprisingly, several recent constructions of Euler systems do not seem to fit the predictions of this conjecture, and we formulate a more general conjecture which explains these extra objects. The novel aspect of our conjecture is that it predicts that there should often be Euler systems of several different ranks associated to a given Galois representation, and we describe how we expect these objects to be related.