Suppose $K/\mathbb{ Q}_p$ is finite and $\overline{ r}\colon G_K\to \mathrm{ GL}_2(\overline{ \mathbb{ F}}_p)$ is a reducible Galois representation. In this paper we prove that we can use the results by the author in [Ste22] to obtain a decomposition of the set of Serre weights $W(\overline{ r})$ into a disjoint union of at most $(e+1)^f$ 'packets' of weights (where $f$ is the residue degree and $e$ the ramification degree of $K$) under the assumption that $\overline{ r}$ is weakly generic. Thereby, we improve on results of Diamond--Savitt in [DS15] which give a similar decomposition, by rather different methods, under the assumption that $\overline{ r}$ is strongly generic. We show that our definition of weak genericity is optimal for the results of this paper to hold when $e=1$. However, we expect that for $e=2$ one of the main results of this paper still holds under weaker hypotheses than the ones used in this paper.
Categories: math.NT