Presentations of Galois groups of unramified extensions of global fields and its predicted distribution
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914564997644288 |
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| author | Willyard, Ken |
| author_facet | Willyard, Ken |
| contents | Motivated by the work of Liu, we study certain canonical quotients of $G_{\emptyset}^T(K)$ -- the Galois group of the maximal unramified extension of a global field $K$ that is split completely at a finite nonempty set of places in $T$ -- for $Γ$-extensions $K/Q$, and prove they have presentations of a particular form. This presentation leads us to the construction of a new random group model as in the work of Liu, Wood, and Zureick-Brown that predicts the distribution of $G_{\emptyset}^T(K)$ as we vary among $Γ$-extensions $K/Q$ with prescribed local conditions at places in $T$, giving a generalization of the non-abelian Cohen-Lenstra-Martinet Heuristics. The key generalization is that $Q$ can be an arbitrary global field, while this comes at a cost of introducing a prime-to-$|\text{Cl}_T(Q)|$ condition in addition to avoiding roots of unity, $|Γ|$, and the characteristic if $Q$ is a function field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14158 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Presentations of Galois groups of unramified extensions of global fields and its predicted distribution Willyard, Ken Number Theory Motivated by the work of Liu, we study certain canonical quotients of $G_{\emptyset}^T(K)$ -- the Galois group of the maximal unramified extension of a global field $K$ that is split completely at a finite nonempty set of places in $T$ -- for $Γ$-extensions $K/Q$, and prove they have presentations of a particular form. This presentation leads us to the construction of a new random group model as in the work of Liu, Wood, and Zureick-Brown that predicts the distribution of $G_{\emptyset}^T(K)$ as we vary among $Γ$-extensions $K/Q$ with prescribed local conditions at places in $T$, giving a generalization of the non-abelian Cohen-Lenstra-Martinet Heuristics. The key generalization is that $Q$ can be an arbitrary global field, while this comes at a cost of introducing a prime-to-$|\text{Cl}_T(Q)|$ condition in addition to avoiding roots of unity, $|Γ|$, and the characteristic if $Q$ is a function field. |
| title | Presentations of Galois groups of unramified extensions of global fields and its predicted distribution |
| topic | Number Theory |
| url | https://arxiv.org/abs/2605.14158 |