We construct an explicit sequence of crystalline representations converging to a given irreducible two-dimensional semi-stable representation of the Gaois group of Q_p. The convergence takes place in the blow-up space of two-dimensional trianguline representations studied by Colmez and Chenevier and is connected to a classical formula due to Greenberg and Stevens expressing the L-invariant as a logarithmic derivative.
Our convergence result can be used to compute the reductions of any irreducible two-dimensional semi-stable representation in terms of the reductions of certain nearby crystalline representations of exceptional weight.
In particular, this provides an alternative approach to computing the reductions of irreducible two-dimensional semi-stable representations that circumvents the somewhat technical machinery of integral p-adic Hodge theory. For instance, using our zig-zag conjecture on the reductions of crystalline representations of exceptional weights, we recover completely the work of Breuil-Mezard and Guerberoff-Park on the reductions of irreducible semi-stable representations of weights at most p+1, at least on the inertia subgroup. As new cases of the zig-zag conjecture are proved, we further obtain some new information about the reductions for small odd weights.
Finally, we use the above ideas to explain away some apparent violations to local constancy in the weight of the reductions of crystalline representations of small weight that were noted in our earlier work and which provided the initial impetus for this work.
This is joint work with Anand Chitrao and Seidai Yasuda.