Abstract
Abstract Let ℓ ≥ 5 {\ell\geq 5} be a prime number and let 𝔽 ℓ {\mathbb{F}_{\ell}} denote the finite field with ℓ {\ell} elements. We show that the number of Galois extensions of the rationals with Galois group isomorphic to GL 2 ( 𝔽 ℓ ) {\operatorname{GL}_{2}(\mathbb{F}_{\ell})} and absolute discriminant bounded above by X is asymptotically at least X ℓ / ( 12 ( ℓ - 1 ) # GL 2 ( 𝔽 ℓ ) ) log X {\frac{X^{{\ell}/({12(\ell-1)\#\operatorname{GL}_{2}(\mathbb{F}_{\ell})})}}{% \log X}} . We also obtain a similar result for the number of surjective homomorphisms ρ : Gal ( ℚ ¯ / ℚ ) → GL 2 ( 𝔽 ℓ ) {\rho:\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{% GL}_{2}(\mathbb{F}_{\ell})} ordered by the prime to ℓ {\ell} part of the Artin conductor of ρ.
Published Version
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