Towards a Modulo $p$ Langlands Correspondence for GL$_2$

The authors construct new families of smooth admissible $/overline{/mathbb{F}}_p$-representations of $/mathrm{GL}_2(F)$, where $F$ is a finite extension of $/mathbb{Q}_p$. When $F$ is unramified, these representations have the $/mathrm{GL}_2({/mathcal O}_F)$-socle predicted by the recent generalizations of Serre's modularity conjecture. The authors' motivation is a hypothetical mod $p$ Langlands correspondence.