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

TL;DR

This paper constructs new infinite families of irreducible supersingular mod p representations of GL_2(F) for ramified p-adic fields, aligning with Serre's conjecture, and provides the first such examples for ramified extensions.

Contribution

It introduces the first known examples of irreducible supersingular representations of GL_2(F) for ramified p-adic fields with specific ramification degrees.

Findings

Constructed infinite families of supersingular representations.

Demonstrated compatibility with Serre's modularity conjecture.

Provided the first examples for ramified extensions.

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 $F / \mathbb{Q}_p$.