Packets of Serre weights for generic locally reducible two-dimensional Galois representations

TL;DR

This paper decomposes the set of Serre weights for certain reducible Galois representations into a limited number of packets, improving previous results by relaxing genericity assumptions and employing new methods.

Contribution

It introduces a decomposition of Serre weights into packets for weakly generic reducible Galois representations, extending prior work that required stronger genericity conditions.

Findings

Decomposition of Serre weights into at most (e+1)^f packets.

Improvement over previous results by Diamond--Savitt.

Optimality of weak genericity condition for e=1.

Abstract

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.