On flagged framed deformation problems of local crystalline Galois representations

Abstract

We prove that irreducible residual crystalline representations of the absolute Galois group of an unramified extension of Qp have smooth representable crystalline framed deformation problems, provided that the Hodge–Tate weights lie in the Fontaine–Laffaille range. We then extend this result to the flagged lifting problem associated to any Fontaine–Laffaille upper triangular representation whose flag is of maximal length. We calculate the relative dimension of these various crystalline lifting functors in terms of the underlying Hodge–Tate weight structures, and also apply these results to give an alternative proof of the fact that every such residual representation admits a so-called “universally twistable lift”. Finally we give some brief indications as to the various directions in which these results might be generalised.