Weak local-global compatibility and ordinary representations

TL;DR

This paper develops a formal framework to prove the pro-modular Fontaine-Mazur conjecture, verifying it in the ordinary case using recent constructions, advancing understanding of local-global compatibility in number theory.

Contribution

It introduces a minimal formalism for proving the pro-modular Fontaine-Mazur conjecture and applies it to the ordinary case with new constructions.

Findings

Proves the pro-modular Fontaine-Mazur conjecture in the ordinary case.

Establishes a general formalism for local-global compatibility.

Utilizes recent constructions by Breuil and Herzig.

Abstract

We introduce a general formalism with minimal requirements under which we are able to prove the pro-modular Fontaine-Mazur conjecture. We verify it in the ordinary case using the recent construction of Breuil and Herzig.