Combinatorial Construction of the Absolute Galois Group of the Field of Rational Numbers
Vol. 32 (2025), No. 1, Page 1-125.
Hoshi, Yuichiro; Mochizuki, Shinichi; Tsujimura, Shota
Combinatorial Construction of the Absolute Galois Group of the Field of Rational Numbers
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Abstract:
In this paper, we give a $\textit {purely combinatorial/group-theoretic construction} $ of the conjugacy class of subgroups of the Grothendieck-Teichmüller group GT determined by the $\textit {absolute Galois group} \; G_\mathbb{Q} \overset{\mathrm{def}}{=}$ Gal $(\bar{\mathbb{Q}}/\mathbb{Q})$ [where $\bar{\mathbb{Q}}$ denotes the field of algebraic numbers] of the field of rational numbers $\mathbb{Q}$. In fact, this construction also yields, as a by-product, a $\textit {purely combinatorial/group-theoretic characterization} $ of the GT-conjugates of $\textit {closed subgroups of}$ $G_\mathbb{Q}$ that are $``\textit {sufficiently large}" $ in a certain sense. We then introduce the notions of $\textit{TKND-fields}$ [i.e., $``\textrm {torally Kummer-nondegenerate fields}"$] and $ \textit {AVKF-fields} $ [i.e., $``\textrm {abelian variety Kummer-faithful fields}"$], which generalize, respectively, the notions of $``$torally Kummer-faithful fields$"$ and $``$Kummer-faithful fields$"$ [notions that appear in previous work of Mochizuki]. For instance, if we write $\mathbb{Q}^{ab} \subseteq \bar{\mathbb{Q}}$ for the maximal abelian extension field of $\mathbb{Q}$, then every finite extension of $\mathbb{Q}^{ab}$ is a ${\it TKND\textrm{-}AVKF\text{-}field}$ [i.e., both TKND and AVKF]. We then apply the purely combinatorial/group-theoretic characterization referred to above to prove that, if a subfield $K \subseteq \bar{\mathbb{Q}}$ is TKND-AVKF, then the commensurator in GT of the subgroup $G_K\subseteq G_\mathbb{Q}$ determined by $K$ is contained in $G_\mathbb{Q}$. Finally, we combine this computation of the commensurator with a result of Hoshi-Minamide-Mochizuki concerning GT to prove a $\textit{semi-absolute version of the Grothendieck Conjecture}$ for higher dimensional [i.e., of dimension $\geq 2$] configuration spaces associated to hyperbolic curves of genus zero over TKND-AVKF-fields.
Keywords: anabelian geometry, étale fundamental group, Grothendieck-Teichmüller group, hyperbolic curve, configuration space, combinatorial Belyi cuspidalization, Grothendieck Conjecture.
Mathematics Subject Classification (2020): Primary 14H30; Secondary 14H25.
Received: 2020-12-14