Claus Sorensen (Prrinceton)
Eigenvarieties and the Breuil-Schneider conjecture
The Fontaine-Mazur conjecture predicts which p-adic Galois representations arise geometrically. A few years back, Emerton and Kisin made astounding progress in the two-dimensional case, by employing the p-adic Langlands correspondence for GL(2) over Q_p, which is very well-understood by now. A key point was the existence of "locally algebraic" vectors. For groups of higher rank, even a conjectural generalization remains elusive. However, there is a conjecture of Breuil
and Schneider, which gives a weak (but precise) analogue for GL(n). Roughly it says that a certain filtration exists if and only if a certain lattice exists. In his thesis, Hu completely proved one direction, and produced the expected filtration (by translating its existence into the so-called Emerton condition). We will report on progress in the other direction, and in many cases prove the existence of GL(n)-stable lattices in locally algebraic representations constructed from p-adic Hodge theoretical data. This argument is global in nature; the ultimate integral structure comes from p-adic modular forms. We hope to also hint at a formalism, in which an eigenvariety for U(n) parametrizes a correspondence between semisimple Galois representations and Banach-Hecke modules with a unitary GL(n)-action, and discuss local-global compatibility "at p" in this context. In particular, we'll settle the Breuil-Schneider conjecture for dR representations which "come from an eigenvariety".