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

  • Anwesh Ray ORCID logo EMAIL logo
Published/Copyright: January 13, 2025
Forum Mathematicum
From the journal Forum Mathematicum Volume 37 Issue 5

Abstract

Let β„“ β‰₯ 5 be a prime number and let 𝔽 β„“ denote the finite field with β„“ elements. We show that the number of Galois extensions of the rationals with Galois group isomorphic to GL 2 ⁑ ( 𝔽 β„“ ) and absolute discriminant bounded above by X is asymptotically at least X β„“ / ( 12 ⁒ ( β„“ - 1 ) ⁒ # ⁒ GL 2 ⁑ ( 𝔽 β„“ ) ) log ⁑ X . We also obtain a similar result for the number of surjective homomorphisms ρ : Gal ⁑ ( β„š Β― / β„š ) β†’ GL 2 ⁑ ( 𝔽 β„“ ) ordered by the prime to β„“ 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

Β© 2025 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Rings of differential operators on (k,s)-th Tjurina algebras of singularities
  3. Cokernels of random matrix products and flag Cohen–Lenstra heuristic
  4. Gradient bounds and Liouville property for a class of hypoelliptic diffusion via coupling
  5. The L p -L q compactness of commutators of oscillatory singular integrals
  6. Determination of a pair of newforms from the product of their twisted central values
  7. Metrical properties of exponentially growing partial quotients
  8. Nonlinear operations and factorizations on a class of affine modulation spaces
  9. On algebraic degrees of certain exponential sums over finite fields
  10. The ranks of (a,b)-Fibonacci sequences and congruences for certain partition functions and Ramanujan's mock theta functions
  11. Dynamics of radial threshold solutions for generalized energy-critical Hartree equation
  12. Modular representations of GL2(π”½π‘ž) using calculus
  13. Bounding the number of p'-degree characters from below
  14. Existence and multiplicity of non-radial sign-changing solutions for a semilinear elliptic equation in hyperbolic space
  15. Duality theorems for polyanalytic functions
  16. Lower bounds for the number of number fields with Galois group GL2(𝔽ℓ)
  17. Cylindrical ample divisors on Du Val del Pezzo surfaces
https://doi.org/10.1515/forum-2024-0059
Keywords for this article
Number field counting; sieve methods; Galois images associated to elliptic curves

Articles in the same Issue

  1. Frontmatter
  2. Rings of differential operators on (k,s)-th Tjurina algebras of singularities
  3. Cokernels of random matrix products and flag Cohen–Lenstra heuristic
  4. Gradient bounds and Liouville property for a class of hypoelliptic diffusion via coupling
  5. The L p -L q compactness of commutators of oscillatory singular integrals
  6. Determination of a pair of newforms from the product of their twisted central values
  7. Metrical properties of exponentially growing partial quotients
  8. Nonlinear operations and factorizations on a class of affine modulation spaces
  9. On algebraic degrees of certain exponential sums over finite fields
  10. The ranks of (a,b)-Fibonacci sequences and congruences for certain partition functions and Ramanujan's mock theta functions
  11. Dynamics of radial threshold solutions for generalized energy-critical Hartree equation
  12. Modular representations of GL2(π”½π‘ž) using calculus
  13. Bounding the number of p'-degree characters from below
  14. Existence and multiplicity of non-radial sign-changing solutions for a semilinear elliptic equation in hyperbolic space
  15. Duality theorems for polyanalytic functions
  16. Lower bounds for the number of number fields with Galois group GL2(𝔽ℓ)
  17. Cylindrical ample divisors on Du Val del Pezzo surfaces
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