Reductions of Galois representations of small slopes

We investigate the shape of the reduction of certain crystalline Galois representations of integral slope 1 and of fractional slopes in (1,2). The proof uses the compatibility between the p-adic and mod p Local Langlands Correspondences with respect to the process of reduction. We give an essentially complete description of the reduction, and show that in the former case it is generically reducible, whereas in the latter case it is generically irreducible. This is joint work with Shalini Bhattacharya for slopes in (1,2), and also with Sandra Rozensztajn for slope 1.