{"paper":{"title":"Presentations of Galois groups of unramified extensions of global fields and its predicted distribution","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Canonical quotients of Galois groups for unramified extensions over global fields have presentations enabling a new random model for their distributions.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ken Willyard","submitted_at":"2026-05-13T22:22:32Z","abstract_excerpt":"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 $\\Gamma$-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 $\\Gamma$-extensions $K/Q$ with prescribed local conditions at places in $T$, giv"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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_∅^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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the proven presentations of the canonical quotients are of a form that directly permits the same random-model construction used by Liu, Wood, and Zureick-Brown, and that the added prime-to-|Cl_T(Q)| condition suffices to make the model work for arbitrary global fields Q.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Establishes group presentations for quotients of G_∅^T(K) in Γ-extensions and derives a random model predicting the distribution of these Galois groups over arbitrary global base fields Q.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Canonical quotients of Galois groups for unramified extensions over global fields have presentations enabling a new random model for their distributions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"ebda571fb3a5b6af8ce259d074ca0a39166d2a30da7d29a84767ce9a8c2bd6a8"},"source":{"id":"2605.14158","kind":"arxiv","version":1},"verdict":{"id":"081748e6-9528-464e-b8b8-ea2bd3d96748","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:48:49.716758Z","strongest_claim":"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_∅^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.","one_line_summary":"Establishes group presentations for quotients of G_∅^T(K) in Γ-extensions and derives a random model predicting the distribution of these Galois groups over arbitrary global base fields Q.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the proven presentations of the canonical quotients are of a form that directly permits the same random-model construction used by Liu, Wood, and Zureick-Brown, and that the added prime-to-|Cl_T(Q)| condition suffices to make the model work for arbitrary global fields Q.","pith_extraction_headline":"Canonical quotients of Galois groups for unramified extensions over global fields have presentations enabling a new random model for their distributions."},"references":{"count":41,"sample":[{"doi":"","year":null,"title":"Cohomology of","work_id":"a96e9000-3bd3-439c-a904-392e7927d976","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Inventiones mathematicae , author =","work_id":"6af899cf-958e-4630-a677-8d973af133f4","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Algebra & Number Theory , author =","work_id":"1014084c-e2d0-4956-9ee4-9b61ea2e94bb","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/s10801-015-0598-x","year":null,"title":"Clancy, Julien and Kaplan, Nathan and Leake, Timothy and Payne, Sam and Wood, Melanie Matchett , year =. 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