Existence of invariant norms in p-adic representations of GL2(F) of large weights

Let F be a finite extension of Q(p) and let q be the cardinality of its residue field. The Breuil-Schneider conjecture for G = GL(n)(F)[BS07] gives a necessary and sufficient condition for the existence of an invariant norm on rho circle times pi, where rho is an irreducible algebraic representation of G and pi is an irreducible smooth representation of G over (F) over bar. The conjecture is still open, even when n = 2, if pi is a principal series representation. In this case, assuming pi is unramified and rho = Sym(k) circle times det(m), it had been verified by Breuil [Bre03b] and De Ieso [DI13] when k < q. We extend their work to the range k < q(2)/2, imposing some technical conditions on pi. (c) 2021 Elsevier Inc. All rights reserved.