Florian Herzig

Weights in a Serre-type conjecture for n-dimensional Galois representations

We discuss a generalisation of Serre's Conjecture to n-dimensional mod p Galois representations ρ that are tamely ramified at p, improving a previous conjecture of Ash, Doud, Pollack, and Sinnott. The weights are predicted in a representation-theoretic way, in terms of the reduction mod p of a Deligne-Lusztig representation associated to ρ. Theoretical and computational evidence will be discussed. We have moreover formulated a Serre-type Conjecture for GSp4 in joint work with J.~Tilouine, whose recent companion forms result in this context provides evidence for some of the predicted weights.