Explicit Serre Weight Conjectures in Dimension Four

Abstract

A generalization of the weight part of Serre's conjecture asks for which Serre weights a given mod p representation of the absolute Galois group of Q is modular. This set is expected to depend only on the restriction of the representation to the Galois group of Q_p. Let rho be a continuous representation of the absolute Galois group of Q_p into GL_n(F_p) that is moreover semisimple. Gee, Herzig, and Savitt [GHS16] defined a certain set W_expl(rho) of Serre weights (which is defined in a very explicit way) that is conjectured to be the correct set of Serre weights as long as rho is sufficiently generic.However, in the non-generic cases that occur in dimensions greater than three, it is not known whether this set behaves in the way it should under certain functorial operations, like tensor products. This thesis shows that in dimension four, the set of explicit Serre weights W_expl(rho) defined in [GHS16] is closed under taking tensor products of two two-dimensional representations.