# Sur un probl\`eme de compatibilit\'e local-global localement analytique

## TL;DR

This paper refines a conjecture by Breuil on locally analytic Ext^1 groups using $(,
)$-modules over the Robba ring and proves several special cases, including for ${
m GL}_3(Q_p)$.

## Contribution

It provides a more precise formulation of Breuil's conjecture and proves new cases, advancing understanding of local-global compatibility in p-adic representation theory.

## Key findings

- Proved special cases of the refined conjecture for ${
m GL}_3(Q_p)$.
- Reinterpreted Breuil's conjecture using $(,
)$-modules over the Robba ring.
- Enhanced the understanding of local-global compatibility in the context of p-adic groups.

## Abstract

We reinterpret a conjecture of Breuil on the locally analytic $\mathrm{Ext}^1$ in a functorial way using $(\varphi,\Gamma)$-modules (possibly with $t$-torsion) over the Robba ring, making it more accurate. Then we prove several special or partial cases of this "improved" conjecture, notably for ${\rm GL}_3(\mathbb{Q}_p)$.

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Source: https://tomesphere.com/paper/1902.03357