Towards a proof of the zig-zag conjecture

2022-11-01   (16:00 ~ 17:00)
Building 108, Room 318
  • Speaker: Eknath Ghate (TIFR)

  • Title:Towards a proof of the zig-zag conjecture

  • Abstract:

The zig-zag conjecture predicts that the reductions of two-dimensional crystalline representations of the Galois group of Q_p of exceptional weights vary essentially through an alternating sequence of explicit irreducible and reducible representations.

Here exceptional means that the slope is a half-integer and the weight is two more than twice the slope modulo (p-1). This conjecture is known to be true for some small slopes: slope 1/2 (Buzzard-Gee), slope 1 (Bhattacharya-Ghate-Rozenstajn), and slope 3/2 (Ghate-Rai).

In this talk we shall give a proof of the zig-zag conjecture on the inertia subgroup, for half-integral slopes bounded by (p-1)/2, for weights that are p-adically close to weights bounded by p+1. The proof uses a recent limiting argument of Ghate-Chitrao-Yasuda connecting two-dimensional crystalline and semi-stable representations, and work of Breuil-Mezard and Guerberoff-Park on the reductions of semi-stable representations of weights bounded by p+1.