Euler systems for Hilbert modular surfaces

TL;DR

This paper constructs an Euler system for Galois representations associated with Hilbert modular surfaces, providing tools to approach the Iwasawa main conjecture under certain conjectural assumptions.

Contribution

It introduces a new Euler system for Galois representations related to Hilbert modular surfaces, advancing the understanding of their arithmetic properties.

Findings

Constructed an Euler system for these Galois representations.

Provides bounds towards the Iwasawa main conjecture.

Links the non-triviality of the Euler system to Bloch-Kato conjecture.

Abstract

We construct an Euler system -- a compatible family of global cohomology classes -- for the Galois representations appearing in the geometry of Hilbert modular surfaces. If a conjecture of Bloch and Kato on injectivity of regulator maps holds, this Euler system is non-trivial, and we deduce bounds towards the Iwasawa main conjecture for these Galois representations.