Reductions of Galois Representations of Exceptional Weights

We describe the reductions of certain two-dimensional Galois representations of small slopes, using the compatibility with respect to reduction between the p-adic and mod p Local Langlands Correspondences. We concentrate on the most interesting and difficult case of weights which are exceptional for a particular half-integral slope. These are weights which are congruent to two more than twice the slope mod (p-1). We make a Zig-Zag Conjecture describing the reduction for such weights in general, and describe some recent evidence towards it.