Anabelian presentation of the motivic Galois group in characteristic zero

Abstract

Let k be a field endowed with a complex embedding, and let G mot 𝒢_mot script upper G Subscript mot be the associated motivic Galois group. The goal of this paper is to show that G mot 𝒢_mot script upper G Subscript mot admits a natural manifestation in anabelian geometry. Roughly speaking, we prove that G mot 𝒢_mot script upper G Subscript mot coincides with the automorphism group of the functor X ↦ π 1 geo ( X ) X↦ π ₁^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.

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BibTeX RIS

BibTeX

@article{ayoub2026anabelian,
  title = {{Anabelian presentation of the motivic Galois group in characteristic zero}},
  author = {Joseph Ayoub},
  journal = {Forum of Mathematics, Sigma},
  year = {2026},
  month = oct,
  doi = {10.1017/fms.2026.10289},
  url = {https://www.cambridge.org/core/product/identifier/S2050509426102898/type/journal_article},
  note = {CC BY 4.0},
}

RIS

TY  - JOUR
TI  - Anabelian presentation of the motivic Galois group in characteristic zero
AU  - Joseph Ayoub
PY  - 2026
DA  - 2026/10/27
JO  - Forum of Mathematics, Sigma
DO  - 10.1017/fms.2026.10289
UR  - https://www.cambridge.org/core/product/identifier/S2050509426102898/type/journal_article
AB  - Let k be a field endowed with a complex embedding, and let G mot 𝒢_mot script upper G Subscript mot be the associated motivic Galois group. The goal of this paper is to show that G mot 𝒢_mot script upper G Subscript mot admits a natural manifestation in anabelian geometry. Roughly speaking, we prove that G mot 𝒢_mot script upper G Subscript mot coincides with the automorphism group of the functor X ↦ π 1 geo ( X ) X↦ π _1^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.
SN  - 2050-5094
ER  -

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