The p-adic Langlands program, pioneered by Breuil and his collaborators, envisions a link between p-adic Hodge theory (which studies Galois representations coming from geometry) and non-archimedean functional analysis. (More precisely, unitary representations of p-adic Lie groups on p-adic Banach spaces.) This correspondence is in fairly good shape for GL(2) over Q_p. For GL(n), there is a conjecture of Breuil and Schneider, which is a somewhat crude (but precise) approximation to what one expects (which is still vague). We will report on our proof of this conjecture in the Steinberg case (in fact, we prove the analogue for discrete series on any reductive group). This is done by passing to a global setting, using trace formula techniques, and carving out a Banach space of classical p-adic modular forms.