Enumerating Galois extensions of number fields
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916277855977472 |
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| author | Oliver, Robert J. Lemke |
| author_facet | Oliver, Robert J. Lemke |
| contents | Let $k$ be a number field. We provide an asymptotic formula for the number of Galois extensions of $k$ with absolute discriminant bounded by some $X \geq 1$, as $X\to\infty$. We also provide an asymptotic formula for the closely related count of extensions $K/k$ whose normal closure has discriminant bounded by $X$. The key behind these results is a new upper bound on the number of Galois extensions of $k$ with a given Galois group $G$ and discriminant bounded by $X$; we show the number of such extensions is $O_{[k:\mathbb{Q}],G} (X^{ \frac{4}{\sqrt{|G|}}})$. This improves over the previous best bound $O_{k,G,ε}(X^{\frac{3}{8}+ε})$ due to Ellenberg and Venkatesh. In particular, ours is the first bound for general $G$ with an exponent that decays as $|G| \to \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04033 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Enumerating Galois extensions of number fields Oliver, Robert J. Lemke Number Theory Group Theory Let $k$ be a number field. We provide an asymptotic formula for the number of Galois extensions of $k$ with absolute discriminant bounded by some $X \geq 1$, as $X\to\infty$. We also provide an asymptotic formula for the closely related count of extensions $K/k$ whose normal closure has discriminant bounded by $X$. The key behind these results is a new upper bound on the number of Galois extensions of $k$ with a given Galois group $G$ and discriminant bounded by $X$; we show the number of such extensions is $O_{[k:\mathbb{Q}],G} (X^{ \frac{4}{\sqrt{|G|}}})$. This improves over the previous best bound $O_{k,G,ε}(X^{\frac{3}{8}+ε})$ due to Ellenberg and Venkatesh. In particular, ours is the first bound for general $G$ with an exponent that decays as $|G| \to \infty$. |
| title | Enumerating Galois extensions of number fields |
| topic | Number Theory Group Theory |
| url | https://arxiv.org/abs/2406.04033 |