Counting $C_2 \wr S_4$ fields with a power saving error term
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914220184961024 |
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| author | Bose, Sambhabi McGown, Kevin J. Panpaliya, Ishan Welling, Natalie Williams, Laney |
| author_facet | Bose, Sambhabi McGown, Kevin J. Panpaliya, Ishan Welling, Natalie Williams, Laney |
| contents | Let $N_d(G,X)$ denote the number of degree $d$ extensions of $\mathbb{Q}$ with Galois closure $G$ and $|Δ_K|\leq X$. Malle's conjecture predicts an asymptotic of the form $N_d(G,X)\sim CX^α(\log X)^β$. Previously, Klüners proved Malle's conjecture for $G=C_2 \wr S_4$. His proof gives a power savings of $O(X^{7/8})$. We improve Klüners' result by establishing a stronger power saving error term for the count of such fields. Specifically, we show $N_8(C_2\wr S_4,X)=CX+O(X^{3/4-1/30})$. Additionally, we obtain new bounds on $N_8(G,X)$ for the groups $S_4$, $C_2^3 \rtimes S_4$, $GL_2 (\mathbb{F}_3)$, and $Q_8\rtimes S_4$ as permutation subgroups of $S_8$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21427 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting $C_2 \wr S_4$ fields with a power saving error term Bose, Sambhabi McGown, Kevin J. Panpaliya, Ishan Welling, Natalie Williams, Laney Number Theory Primary: 11R45, 11N45, Secondary: 11R29 Let $N_d(G,X)$ denote the number of degree $d$ extensions of $\mathbb{Q}$ with Galois closure $G$ and $|Δ_K|\leq X$. Malle's conjecture predicts an asymptotic of the form $N_d(G,X)\sim CX^α(\log X)^β$. Previously, Klüners proved Malle's conjecture for $G=C_2 \wr S_4$. His proof gives a power savings of $O(X^{7/8})$. We improve Klüners' result by establishing a stronger power saving error term for the count of such fields. Specifically, we show $N_8(C_2\wr S_4,X)=CX+O(X^{3/4-1/30})$. Additionally, we obtain new bounds on $N_8(G,X)$ for the groups $S_4$, $C_2^3 \rtimes S_4$, $GL_2 (\mathbb{F}_3)$, and $Q_8\rtimes S_4$ as permutation subgroups of $S_8$. |
| title | Counting $C_2 \wr S_4$ fields with a power saving error term |
| topic | Number Theory Primary: 11R45, 11N45, Secondary: 11R29 |
| url | https://arxiv.org/abs/2512.21427 |