  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 $\R$-anisotropic model of $G$. The ultimate norm
comes from classical $p$-adic modular forms.
