"Integral structures in Steinberg representations and p-adic Langlands"
Abstract: 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. For the most part, the colloquium will be very low-key and widely accessible. Tea at 4:15
Friday January 25, 2013 at 3:00 PM in SEO 636