Towards a modulo 𝑝 Langlands correspondence for GL₂

We construct new families of smooth admissible F ¯ p F̄_p -representations of G L 2 ( F ) GL_2(F) , where F F is a finite extension of Q p Q_p . When F F is unramified, these representations have the G L 2 ( O F ) GL_2({ O}_F) -socle predicted by the recent generalizations of Serre’s modularity conjecture. Our motivation is a hypothetical mod p p Langlands correspondence.