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On unipotent radicals of motivic Galois groups

Payman Eskandari and V. Kumar Murty

Vol. 17 (2023), No. 1, 165–215

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 𝔲¯(M) W1Hom ¯(M,M) the Lie algebra of the kernel of the natural surjection from the fundamental group of M to the fundamental group of Gr WM. A result of Deligne gives a characterization of 𝔲¯(M) in terms of the extensions 0 WpM M MWpM 0: it states that 𝔲¯(M) is the smallest subobject of W1Hom ¯(M,M) such that the sum of the aforementioned extensions, considered as extensions of by W1Hom ¯(M,M), is the pushforward of an extension of by 𝔲¯(M). We study each of the above-mentioned extensions individually in relation to 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 WpM M MWpM 0 is the pushforward of an extension of by 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 𝔲¯(M) = W1Hom ¯(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 with three weights and large unipotent  radicals.

mixed motives, motivic Galois groups, periods
Mathematical Subject Classification
Primary: 14F42
Secondary: 11M32, 18M25, 32G20
Received: 19 October 2021
Revised: 25 January 2022
Accepted: 4 March 2022
Published: 24 March 2023
Payman Eskandari
Department of Mathematics and Statistics
University of Winnipeg
V. Kumar Murty
Department of Mathematics
University of Toronto
Toronto ON

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