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.