Number Theory Seminar: Zig-Zag holds for Galois representations

Describing the shape of the reduction of local Galois representations attached to cusp forms, at primes away from the level — or more generally, the reduction of two-dimensional crystalline representations — remains an open problem. Partial results go back to Deligne, Fontaine, and Edixhoven. One folklore conjecture attributed to Breuil, Buzzard and Emerton predicts that the reduction is irreducible if the weight is even and the slope is fractional.

In this talk we state and prove the zig-zag conjecture, which treats the case of large exceptional weights and half-integral slopes. The conjecture predicts that the reduction alternates between certain irreducible and reducible representations, in a pattern governed by the relative size of two auxiliary parameters. Special cases of zig-zag have previously been established using Langlands correspondences.

Our proof reverses a recent limiting argument of Chitrao–Ghate–Yasuda in the Colmez–Chenevier rigid-analytic space of trianguline representations, reducing the crystalline case to results on the reduction of semistable representations due to Breuil–Mézard, Guerberoff–Park, and more recently, Chitrao–Ghate.