On the existence of large degree Galois representations for fields of small discriminant

TL;DR

This paper establishes lower bounds on the degrees of irreducible Galois representations for number fields with small discriminant, suggesting such fields cannot have overly abelian Galois groups, with some results depending on unproven hypotheses.

Contribution

It provides new lower bounds on Galois representation degrees for fields with small discriminant, advancing understanding of the structure of Galois groups in number theory.

Findings

Lower bounds are nontrivial for small degree fields with small root discriminant.

Results are sharper under the assumption of the Artin Conjecture and GRH.

Fields with small discriminant cannot have Galois groups that are too abelian.

Abstract

Let $L/K$ be a Galois extension of number fields. We prove two lower bounds on the maximum of the degrees of the irreducible complex representations of ${\rm Gal}(L/K)$, the sharper of which is conditional on the Artin Conjecture and the Generalized Riemann Hypothesis. Our bound is nontrivial when $[K : \mathbb{Q}]$ is small and $L$ has small root discriminant, and might be summarized as saying that such fields can't be "too abelian."