Sur quelques repr\'esentations supersinguli\`eres de $GL_2(Q_{p^f})$

TL;DR

This paper explores new supersingular representations of GL_2 over unramified extensions of p-adic fields, revealing additional parameters beyond previously known ones, advancing understanding in p-adic representation theory.

Contribution

It demonstrates the existence of more parameters in supersingular representations of GL_2(Q_{p^f}) than previously identified, expanding the classification framework.

Findings

More parameters than known in supersingular representations

Extension of Breuil and Paskunas' parametrization

Enhanced understanding of mod p representations of GL_2(Q_{p^f})

Abstract

Let p>3 be a prime, f a positive integer and Q_{p^f} the unramified extension of Q_p of degree f. After Breuil and Paskunas, to a generic semi-simple continue modulo p representation of the absolute Galois group of Q_{p^f}, we can associate a parameterized family of smooth admissible modulo p representations of GL_2(Q_{p^f}). In this article, we prove that there are more parameters than those known.