Anabelian presentation of the motivic Galois group in characteristic zero
Year: 2026
Type: article
Abstract: Let k be a field endowed with a complex embedding, and let G mot $\mathcal {G}_{\mathrm {mot}}$ script upper G Subscript mot be the associated motivic Galois group. The goal of this paper is to show that G mot $\mathcal {G}_{\mathrm {mot}}$ script upper G Subscript mot admits a natural manifestation in anabelian geometry. Roughly speaking, we prove that G mot $\mathcal {G}_{\mathrm {mot}}$ script upper G Subscript mot coincides with the automorphism group of the functor X ↦ π 1 geo ( X ) $X\mapsto \pi _1^{\mathrm {geo}}(X)$ upper X right arrow from bar pi 1 Superscript geo Baseline left parenthesis upper X right parenthesis sending a k -variety X to the geometric completion of its topological fundamental group. This can be considered as a motivic version of the Ihara–Matsumoto–Oda Conjecture, greatly generalised and proven by Pop. In a sequel to this paper, we will develop the parallel story for the ℓ $\ell $ script l -adic realisation. For this reason, many of the intermediate results are developed at a greater level of generality than strictly necessary for the main results proven here.
Source: Forum of Mathematics Sigma
Author: Joseph Ayoub
Institution: University of Zurich
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $2,650
OpenAlex ID: W7223794291