Lower bounds for the number of number fields with Galois group $GL_2(\mathbb{F}_\ell)$
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913875372277760 |
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| author | Ray, Anwesh |
| author_facet | Ray, Anwesh |
| contents | Let $\ell\geq 5$ be a prime number and $\mathbb{F}_\ell$ denote the finite field with $\ell$ elements. We show that the number of Galois extensions of the rationals with Galois group isomorphic to $GL_2(\mathbb{F}_\ell)$ and absolute discriminant bounded above by $X$ is asymptotically at least $\frac{X^{\frac{\ell}{12(\ell-1)\# GL_2(\mathbb{F}_\ell)}}}{\log X}$. We also obtain a similar result for the number of surjective homomorphisms $ρ:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_2(\mathbb{F}_\ell)$ ordered by the prime to $\ell$ part of the Artin conductor of $ρ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_01956 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lower bounds for the number of number fields with Galois group $GL_2(\mathbb{F}_\ell)$ Ray, Anwesh Number Theory 11R32, 11R45 Let $\ell\geq 5$ be a prime number and $\mathbb{F}_\ell$ denote the finite field with $\ell$ elements. We show that the number of Galois extensions of the rationals with Galois group isomorphic to $GL_2(\mathbb{F}_\ell)$ and absolute discriminant bounded above by $X$ is asymptotically at least $\frac{X^{\frac{\ell}{12(\ell-1)\# GL_2(\mathbb{F}_\ell)}}}{\log X}$. We also obtain a similar result for the number of surjective homomorphisms $ρ:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_2(\mathbb{F}_\ell)$ ordered by the prime to $\ell$ part of the Artin conductor of $ρ$. |
| title | Lower bounds for the number of number fields with Galois group $GL_2(\mathbb{F}_\ell)$ |
| topic | Number Theory 11R32, 11R45 |
| url | https://arxiv.org/abs/2305.01956 |