On unipotent radicals of motivic Galois groups

Eskandari, Payman; Murty, V. Kumar

doi:10.2140/ant.2023.17.165

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On unipotent radicals of motivic Galois groups

Payman Eskandari and V. Kumar Murty

Vol. 17 (2023), No. 1, 165–215

DOI: 10.2140/ant.2023.17.165

Abstract

Let $T$ be a neutral Tannakian category over a field of characteristic zero with unit object $⊮$ , and equipped with a filtration $W_{∙}$ similar to the weight filtration on mixed motives. Let $M$ be an object of $T$ , and $\underline{𝔲} (M) \subset W_{- 1} \underline{Hom} (M, M)$ the Lie algebra of the kernel of the natural surjection from the fundamental group of $M$ to the fundamental group of ${Gr}^{W} M$ . A result of Deligne gives a characterization of $\underline{𝔲} (M)$ in terms of the extensions $0 \to W_{p} M \to M \to M ∕ W_{p} M \to 0$ : it states that $\underline{𝔲} (M)$ is the smallest subobject of $W_{- 1} \underline{Hom} (M, M)$ such that the sum of the aforementioned extensions, considered as extensions of $⊮$ by $W_{- 1} \underline{Hom} (M, M)$ , is the pushforward of an extension of $⊮$ by $\underline{𝔲} (M)$ . We study each of the above-mentioned extensions individually in relation to $\underline{𝔲} (M)$ . Among other things, we obtain a refinement of Deligne’s result, where we give a sufficient condition for when an individual extension $0 \to W_{p} M \to M \to M ∕ W_{p} M \to 0$ is the pushforward of an extension of $⊮$ by $\underline{𝔲} (M)$ . In the second half of the paper, we give an application to mixed motives whose unipotent radical of the motivic Galois group is as large as possible (i.e., with $\underline{𝔲} (M) = W_{- 1} \underline{Hom} (M, M)$ ). Using Grothendieck’s formalism of extensions panachées we prove a classification result for such motives. Specializing to the category of mixed Tate motives we obtain a classification result for 3-dimensional mixed Tate motives over $ℚ$ with three weights and large unipotent radicals.

Keywords

mixed motives, motivic Galois groups, periods

Mathematical Subject Classification

Primary: 14F42

Secondary: 11M32, 18M25, 32G20

Milestones

Received: 19 October 2021

Revised: 25 January 2022

Accepted: 4 March 2022

Published: 24 March 2023

Authors

Payman Eskandari
Department of Mathematics and Statistics University of Winnipeg Winnipeg MB Canada

V. Kumar Murty
Department of Mathematics University of Toronto Toronto ON Canada

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Vol. 17, No. 1, 2023