as a vast generalization of quadratic reciprocity, class field theory describes all abelian extensions of a number field. over q, they are precisely those contained in cyclotomic fields. however, there are a lot more non-abelian extensions, which arise naturally. the langlands program attempts to systematize them, by relating galois representations and automorphic forms; mathematical objects of rather disparate nature. we will illustrate the basic plot for gl(2) through the example of elliptic curves and modular forms - the context of wiles' proof of fermat's last theorem. the main goal of the talk will be to motivate a ``p-adic" langlands correspondence, which is at the forefront of contemporary number theory, but still only well-understood for gl(2) over $q_p$. we will discuss, in some depth, the case of semistable elliptic curves, which provide the first non-trivial example. this leads naturally to a result we proved recently, which shows the existence of (many) integral structures in locally algebraic representations of ``steinberg" type, for any reductive group g (such as gl(n), symplectic, and orthogonal groups). as a result, there are a host of ways to p-adically complete the steinberg representation (tensored with an algebraic representation). the ensuing banach spaces should play a role in a (yet elusive) higher-dimensional p-adic langlands correspondence. we hope to at least give some idea of the proof, which goes via automorphic representations and the trace formula.