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arXiv · 1506.02177

Motivic Serre group, algebraic Sato-Tate group and Sato-Tate conjecture

Grzegorz Banaszak·Kiran S. Kedlaya
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Abstract

We make explicit Serre's generalization of the Sato-Tate conjecture for motives, by expressing the construction in terms of fiber functors from the motivic category of absolute Hodge cycles into a suitable category of Hodge structures of odd weight. This extends the case of abelian varietes, which we treated in a previous paper. That description was used by Fite--Kedlaya--Rotger--Sutherland to classify Sato-Tate groups of abelian surfaces; the present description is used by Fite--Kedlaya--Sutherland to make a similar classification for certain motives of weight 3. We also give conditions under which verification of the Sato-Tate conjecture reduces to the identity connected component of the corresponding Sato-Tate group.

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Grzegorz Banaszak, Kiran S. Kedlaya. 2015-06-06. Motivic Serre group, algebraic Sato-Tate group and Sato-Tate conjecture. https://arxiv.org/abs/1506.02177

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