# Weight Cycling and Serre-type Conjectures

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Florian Herzig, Northwestern University
Fine Hall 214

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 $GL_2$. 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.