The imaginary case of the nonabelian Cohen--Lenstra heuristics
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911081634463744 |
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| author | Liu, Yuan Willyard, Ken |
| author_facet | Liu, Yuan Willyard, Ken |
| contents | For a finite group $Γ$, we study the distribution of the Galois group $G_{\emptyset}^{\#}(K)$ of the maximal unramified extension of $K$ that is split completely at $\infty$ and has degree prime to $|Γ|$ and $\textit{Char}(K)$, as $K$ varies over imaginary $Γ$-extensions of $\mathbb{Q}$ or $\mathbb{F}_q(t)$. In the function field case, we compute the moments of the distribution of $G_{\emptyset}^{\#}(K)$ by counting points on Hurwitz stacks. In order to understand the probability of the distribution, we prove that $G_{\emptyset}^{\#}(K)$ admits presentations of a specific form, then use this presentation to build random groups to simulate the behavior of $G_{\emptyset}^{\#}(K)$, and make the conjecture to predict the distribution using the probability measures of these random groups. Our results provide the imaginary analog of the work of Wood, Zureick-Brown, and the first author on the nonabelian Cohen--Lenstra heuristics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_21558 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The imaginary case of the nonabelian Cohen--Lenstra heuristics Liu, Yuan Willyard, Ken Number Theory For a finite group $Γ$, we study the distribution of the Galois group $G_{\emptyset}^{\#}(K)$ of the maximal unramified extension of $K$ that is split completely at $\infty$ and has degree prime to $|Γ|$ and $\textit{Char}(K)$, as $K$ varies over imaginary $Γ$-extensions of $\mathbb{Q}$ or $\mathbb{F}_q(t)$. In the function field case, we compute the moments of the distribution of $G_{\emptyset}^{\#}(K)$ by counting points on Hurwitz stacks. In order to understand the probability of the distribution, we prove that $G_{\emptyset}^{\#}(K)$ admits presentations of a specific form, then use this presentation to build random groups to simulate the behavior of $G_{\emptyset}^{\#}(K)$, and make the conjecture to predict the distribution using the probability measures of these random groups. Our results provide the imaginary analog of the work of Wood, Zureick-Brown, and the first author on the nonabelian Cohen--Lenstra heuristics. |
| title | The imaginary case of the nonabelian Cohen--Lenstra heuristics |
| topic | Number Theory |
| url | https://arxiv.org/abs/2507.21558 |