Zig-zag is true

Abstract: It is an open problem to describe the shape of the reductions of
local Galois representations attached to cusp forms at primes away from
the level, or more generally, the shape of the reductions of two-dimensional crystalline representations. Partial results go back to Deligne, Fontaine and Edixhoven. One folklore conjecture (attributed to Breuil, Buzzard and Emerton) is that if the weight is even and the slope is fractional, then the reduction is always irreducible.

In this talk we shall state and prove our zig-zag conjecture which deals
with large exceptional weights and half-integral slopes. These weights
fall squarely outside the scope of the above conjecture. The conjecture states that the reduction in these cases is given by an alternating sequence of irreducible and reducible representations depending on the size of two
auxiliary parameters.

Special cases of zig-zag have been proved over the years by various
authors using Langlands correspondences. The present general proof uses
the reverse of a recent limiting argument due to Chitrao-Ghate-Yasuda in the
Colmez-Chenevier rigid analytic blow up space of trianguline
representations to reduce the study of the reduction of crystalline
representations to results on the reductions of semi-stable
representations due to Breuil-Mezard, and more recently, Guerberoff-Park.