Determining the shape of the reductions of two-dimensional crystalline representations is an old problem going back to Deligne, and Fontaine and Edixhoven.
In particular, this reduction can be both irreducible and reducible. A conjecture I made about 5 years ago - the zig-zag conjecture - says that for exceptional weights with respect to half-integral slopes, the reduction varies through an alternating sequence of irreducible and reducible representations depending on the relative sizes of two auxiliary parameters.
In this talk we give a survey of work done on the reduction problem, concentrating on our recent proof of the zig-zag conjecture. | 
