nLab mod p local Langlands correspondence

Idea

The mod $p$ local Langlands correspondence is a version of the local Langlands correspondence where the representations are over a finite field of characteristic $p$, i.e. on one side we have certain $k$-representations of $G(F)$ (for $G$ some reductive group), and on the other we have certain $k$-representations of $\mathrm{Gal}(\overline{F}/F)$, where $F$ is a finite extension of $\mathbb{Q}_{p}$, and $k$ is a finite field of characteristic $p$. It should be compatible with the p-adic local Langlands correspondence.The only case that is known is when $G=\mathrm{GL}_{2}$ and $F=\mathbb{Q}_{p}$.

References

• Christophe Breuil, The emerging p-adic Langlands program, Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010 (pdf)