A family of irreducible supersingular representations of $\mathrm{GL}_2(F)$ for some ramified $p$-adic fields

Abstract

We construct infinite families of irreducible supersingular mod $p$ representations of $\mathrm{GL}_2(F)$ with $\mathrm{GL}_2(\mathcal{O}_F)$-socle compatible with Serre's modularity conjecture, where $F / \mathbb{Q}_p$ is any finite extension with residue field $\mathbb{F}_{p^2}$ and ramification degree $e \leq (p-1)/2$. These are the first such examples for ramified $F / \mathbb{Q}_p$.