Every 7-Dimensional Abelian Variety over the p-adic Numbers has a Reducible $\ell$-adic Galois Representation

TL;DR

This paper investigates the possible dimensions of irreducible $$-adic Galois representations over p-adic fields, showing restrictions on their existence based on prime properties, with a focus on 7-dimensional abelian varieties.

Contribution

It establishes new constraints on the dimensions of irreducible $$-adic Galois representations over p-adic fields, especially excluding certain prime dimensions.

Findings

Non-Sophie Germain primes are not in the set of possible dimensions for certain Galois representations.

The 7-dimensional case of abelian varieties over p-adic fields has a reducible $$-adic Galois representation.

Restrictions depend on the residue characteristic being greater than 3.

Abstract

Let $K$ be a complete, discretely valued field with finite residue field and $G_K$ its absolute Galois group. The subject of this note is the study of the set of positive integers $d$ for which there exists an absolutely irreducible $\ell$-adic representation of $G_K$ of dimension $d$ with rational traces on inertia. Our main result is that non-Sophie Germain primes are not in this set when the residue characteristic of $K$ is $> 3$. The result stated in the title is a special case.