Lower bounds for the number of number fields with Galois group GL₂(\mathbbF_ℓ)

2023/05/03 by Ray, Anwesh
#11R32 #11R45 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2305.01956

Abstract

Let ℓ≥ 5 be a prime number and \mathbbF_ℓ denote the finite field with ℓ elements. We show that the number of Galois extensions of the rationals with Galois group isomorphic to GL₂(\mathbbF_ℓ) and absolute discriminant bounded above by X is asymptotically at least \fracX^\fracℓ12(ℓ-1)# GL₂(\mathbbF_ℓ)log X. We also obtain a similar result for the number of surjective homomorphisms ρ:Gal(ℚ/ℚ)→ GL₂(\mathbbF_ℓ) ordered by the prime to ℓ part of the Artin conductor of ρ.

Related

• Lower bounds for GL₂(\mathbbF_ℓ) number fields
• Enumerating Galois extensions of number fields
• Upper Bounds for the Number of Number Fields with Alternating Galois\n Group
• Counting G-Extensions by Discriminant
• Counting extensions of number fields with Frobenius Galois group