Number/Representation Theory

We seek a $p$-adic Langlands correspondence between a Galois representation $\mathrm{Gal}(\overline K/K) \to \mathrm{GL}_n(\overline{\mathbb Q}_p)$ and an admissible unitary representation of $\mathrm{GL}_n(K)$ on a $p$-adic Banach space. This correspondence is known when $\mathrm{GL}_n(K) = \mathrm{GL}_2(\mathbb Q_p)$, but remains unknown even for $\mathrm{GL}_2(\mathbb Q_{p^f})$. That said, given a $p$-adic Galois representation $\mathrm{Gal}(\overline K/K) \to \mathrm{GL}_2(\overline{\mathbb Q}_p)$, one can construct an admissible unitary representation of $\mathrm{GL}_2(\mathbb Q_{p^f})$ using a global setup. However, it is unclear whether this construction is independent of the global setting. Breuil's lattice conjecture provides evidence for such a claim. Proving the conjecture shows strong local–global compatibility. In the talk, I will explain the motivation behind the conjecture and, time permitting, briefly sketch the proof.