Serre weights for locally reducible two-dimensional Galois representations

TL;DR

This paper advances the understanding of Serre's conjecture by characterizing the predicted weights for reducible two-dimensional Galois representations over totally real fields, especially in generic cases.

Contribution

It proves the exact correspondence between predicted and actual Serre weights for generic reducible local representations, and classifies all possible weight subsets.

Findings

Confirmed the predicted weights match the modular weights in generic cases.

Determined which subsets of weights can occur as predictions for fixed semisimplifications.

Provided a classification of predicted weights as the local representations vary.

Abstract

Let F be a totally real field, and v a place of F dividing an odd prime p. We study the weight part of Serre's conjecture for continuous, totally odd, two-dimensional mod p representations rhobar of the absolute Galois group of F that are reducible locally at v. Let W be the set of predicted Serre weights for the semisimplification of rhobar restricted to the decomposition group at v. We prove that when the local representation is generic, the Serre weights in W for which rhobar is modular are exactly the ones that are predicted (assuming that rhobar is modular). We also determine precisely which subsets of W arise as predicted weights when the local representation varies with fixed generic semisimplification.