Galois representations are fundamental tools with which to study many problems in number theory. They are most famously attached to modular forms and primes p. An outstanding problem regarding them is to determine their reductions modulo p in various settings. In the global setting, the shape of the reduction is connected to congruences for the form.

In the local crystalline setting, a particularly difficult case to treat is that of exceptional weights. My zig-zag conjecture predicts that, in this case, the reduction should be given by an explicit alternating sequence of irreducible and reducible representations.

In this talk, I will state the conjecture and announce a proof on inertia for large weights. I will also give a brief outline of the proof, which uses some delicate limiting arguments in a certain blow-up space to reduce the argument to the shape of the reductions of semi-stable representations.