Deformation of the absolute Galois groups of number fields

2026/07/30 by Takeo Uramoto
Mathematics · #math.NT

paper · pdf

draft; 31 pages

arxiv created 2026/07/30 · arxiv updated 2026/07/31

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.

Citations

• Semi-galois Categories IV: A deformed reciprocity law for Siegel modular functions
• Semi-galois Categories III: Witt vectors by deformations of modular functions
• Semi-galois Categories II: An arithmetic analogue of Christol's theorem
• On arithmetic models and functoriality of Bost-Connes systems. With an appendix by Sergey Neshveyev
• Lambda actions of rings of integers

Related

• Relations in the maximal pro-p quotients of absolute Galois groups
• Admissible groups over number fields
• Isomorphisms of Galois groups of number fields with restricted ramification
• Deformation rings and Hecke algebras in the totally real case
• Combinatorial Techniques in the Galois Theory of p-Extensions