On unipotent radicals of motivic Galois groups

Eskandari, Payman; Murty, V. Kumar

doi:10.2140/ant.2023.17.165

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Form

Policies for Authors

Ethics Statement

ISSN: 1944-7833 (e-only)

ISSN: 1937-0652 (print)

Author Index

To Appear

Other MSP Journals

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: 18 February 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

Vol. 17, No. 1, 2023