A proof of the Breuil-Schneider conjecture in the indecomposable case

Abstract

This paper contains a proof of a conjecture of Breuil and Schneider, on the existence of an invariant norm on any locally algebraic representation of $\GL(n)$, with integral central character, whose smooth part is given by a generalized Steinberg representation. In fact, we prove the analogue for any connected reductive group $G$. This is done by passing to a global setting, using the trace formula for an $\mathbb{R}$-anisotropic model of $G$. The ultimate norm comes from classical $p$-adic modular forms.