Florian Herzig (Northwestern)

Weight Cycling and Serre-type Conjectures

Suppose that rho is a three-dimensional modular mod p Galois representation whose restriction to the decomposition groups at p is irreducible and generic. If rho is modular in some (Serre) weight, then a representation-theoretic argument shows that it also has to be modular in certain other weights. (We can give a short list of possibilities). This goes back to an observation of Buzzard for GL₂. Previously we formulated a Serre-type conjecture on the possible weights of rho. Under the assumption that the weights of rho are contained in the predicted weight set, we apply the above weight cycling argument to show that rho is modular in precisely all the nine predicted weights. This is joint work with Matthew Emerton and Toby Gee.