On the $K$-extensions between Serre weights for unramified $\mathrm{GL}_3$

TL;DR

This paper investigates the extension groups between Serre weights for unramified $ ext{GL}_3$ over local fields, showing that in most cases, these extensions match those over the finite field $ ext{GL}_3( extbf{F}_q)$.

Contribution

It establishes a correspondence between extension groups for $ ext{GL}_3( extbf{O}_L)$ and $ ext{GL}_3( extbf{F}_q)$ for most cases, advancing understanding of Serre weights.

Findings

Extensions between Serre weights for $ ext{GL}_3( extbf{O}_L)$ often coincide with those over $ extbf{F}_q$.

Most cases show the extension groups are the same for both groups.

The result clarifies the structure of Serre weight extensions in unramified $ ext{GL}_3$ contexts.

Abstract

Let $p\geq5$ be a prime number. Let $L$ be a finite unramified extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_L$ and residue field $\mathbb{F}_q$. Given two Serre weights for $\mathrm{GL}_3(\mathbb{F}_q)$, we prove that in most cases the extensions between them for $\mathrm{GL}_3(\mathcal{O}_L)$ modulo the center coincide with their $\mathrm{GL}_3(\mathbb{F}_q)$-extensions.