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

Anwesh Ray

doi:10.1515/forum-2024-0059

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

Anwesh Ray

Veröffentlicht/Copyright: 13. Januar 2025

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Aus der Zeitschrift Forum Mathematicum Band 37 Heft 5

Abstract

Let ℓ ≥ 5 be a prime number and let 𝔽 ℓ denote the finite field with ℓ elements. We show that the number of Galois extensions of the rationals with Galois group isomorphic to GL 2 ⁡ ( 𝔽 ℓ ) and absolute discriminant bounded above by X is asymptotically at least X ℓ / ( 12 ⁢ ( ℓ - 1 ) ⁢ # ⁢ GL 2 ⁡ ( 𝔽 ℓ ) ) log ⁡ X . We also obtain a similar result for the number of surjective homomorphisms ρ : Gal ⁡ ( ℚ ¯ / ℚ ) → GL 2 ⁡ ( 𝔽 ℓ ) ordered by the prime to ℓ part of the Artin conductor of ρ.

Keywords: Number field counting; sieve methods; Galois images associated to elliptic curves

MSC 2020: 11G05; 11R45

Communicated by Jan Bruinier

Acknowledgements

When the project was started, the author was a Simons postdoctoral fellow at the Centre de recherches mathematiques in Montreal, Canada. At this time, the author’s research is supported by the CRM Simons postdoctoral fellowship. He would like to thank Chris Wuthrich for his comments in response to a question he posted on MathOverflow. He would like to thank Robert Lemke-Oliver for pointing out a calculation error in the previous version. Lastly, he would also like to thank the anonymous referee for the excellent report.

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Received: 2024-01-30

Revised: 2024-07-30

Published Online: 2025-01-13

Published in Print: 2025-06-01

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Schlagwörter für diesen Artikel

Number field counting; sieve methods; Galois images associated to elliptic curves