Existence de normes invariantes pour GL_2

TL;DR

This paper proves that for GL_2 over a p-adic field, certain admissible filtrations imply the existence of invariant norms on locally algebraic representations, supporting a conjecture linking representation theory and Galois representations.

Contribution

It establishes a partial proof of Breuil and Schneider's conjecture for the case d=2 under specific conditions, connecting invariant norms with admissible filtrations.

Findings

Existence of admissible filtrations implies invariant norms for GL_2.

Supports conjecture relating invariant norms and de Rham Galois representations.

Provides conditions under which the conjecture holds in the case d=2.

Abstract

Breuil et Schneider formulated a conjecture on the equivalence of the existence of invariant norms on certain locally algebraic representations of GL_d(F) and the existence of certain de Rham representations of Gal(\bar(Q_p)/F)$, where F is a finite extension of Q_p. In this paper we prove that in the case d = 2 and under some conditions, the existence of certains admissible filtrations implies the existence of invariant norms.