Self-dual Artin representations of dimension three (with an appendix by David E. Rohrlich)

TL;DR

This paper proves unconditionally that self-dual Artin representations of dimension three over the rationals are rare, having density zero among all such representations, removing the need for prior conjectural assumptions.

Contribution

It provides the first unconditional proof that self-dual Artin representations of dimension three have density zero among all such representations over , previously known only assuming Malle's Conjecture.

Findings

Self-dual Artin representations of dimension three have density zero.

The result was previously conditional on Malle's Conjecture.

The proof removes the reliance on unproven conjectures.

Abstract

We give an unconditional proof that self-dual Artin representations of $\mathbb{Q}$ of dimension $3$ have density $0$ among all Artin representations of $\mathbb{Q}$ of dimension $3$. Previously this was known under the assumption of Malle's Conjecture.