Authors: Herzig, Florian (2006)We formulate a conjecture generalising the weight in Serre's Conjecture to n-dimensional representations p : Gal(U/Q) -* GLn(IF,) that are tamely ramified at p. A weight in this context is an irreducible representation of GLn(1Fp) over Pp. The conjecture describes the predicted set of weights in terms of the reduction modulo p of a Deligne-Lusztig repre¬sentation of GLn(Fp) which only depends on the restriction of p to the inertia subgroup at p.
When n = 3 a weight conjecture had already been made by Ash, Doud, Pollack and Sinnott. The advantage of our conjecture is that it is more conceptual. It moreover predicts more weights for many representations p. We give computational examples which strongly suggest the existence of these extra weights.
When n = 4 we obtain some theoretica...