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arXiv · 2607.28465

Deformation of the absolute Galois groups of number fields

Takeo Uramoto
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Abstract

The technical goal of this paper is to construct and study profinite monoids DG_K for number fields K such that (1) the unit group of DG_K is isomorphic to the absolute Galois group G_K; (2) the maximal abelian quotient of DG_K is isomorphic to the Deligne-Ribet monoid DR_K; (3) the idempotents of DG_K bijectively correspond to subsets of the set P_K of (finite) primes of K; (4) the maximal Galois groups G_{K, S} with restricted ramifications S all appear as maximal closed subgroups of DG_K at idempotents; (5) all maximal closed subgroups of DG_K at idempotents are of this form. We also discuss the relationship between DG_K and the p-typical Witt vectors of Borger and de Smit, where we will describe the fundamental monoid of the semi-galois category of Lambda-rings of Borger and de Smit in terms of fields of norms of the local field K_p in particular.

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Takeo Uramoto. 2026-07-30. Deformation of the absolute Galois groups of number fields. https://arxiv.org/abs/2607.28465

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