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Thursday, December 3, 2015

**Abstract:** Serre's modularity conjecture (now a Theorem due to Khare-Wintenberger and Kisin) states that every odd irreducible two dimensional mod p representation of the absolute Galois group of Q comes from a modular form. I will begin with an overview of the Serre's original conjecture on modular forms focusing on the weight part of the conjecture. Herzig gave a generalization of the conjecture for n-dimensional Galois representations which predicts the modularity of so-called shadow weights. After briefly describing Herzig's conjecture, I will discuss joint work with D. Le, B. Le Hung, and S. Morra where we prove instances of this conjecture in dimension three.