Chow trace of 1-motives and the Lang-N\'eron groups

TL;DR

This paper extends Chow's results on abelian varieties to 1-motives, establishing the existence of Chow trace and image functors, and explores their relation to Voevodsky's motives and Lang-Néron groups.

Contribution

It introduces the Chow trace and image functors for 1-motives over primary field extensions, generalizing classical results and connecting them to Voevodsky's motivic framework.

Findings

Existence of Chow trace and image functors for 1-motives.

Identification of the Chow trace with the zero-th direct image in Voevodsky's motives.

Relation of the first direct image to the Lang-Néron group.

Abstract

We show that in the case of primary field extensions, the extension of scalars of Deligne $1$-motives admits a left adjoint, called Chow image, and a right adjoint, called Chow trace. This generalizes Chow's results on abelian varieties. Then we study the Chow trace in the framework of Voevodsky's triangulated categories of (\'etale) motives. With respect to the $1$-motivic $t$-structure on the category of Voevodsky's homological $1$-motives, the zero-th direct image of an abelian variety is given by the Chow trace, and the first direct image is the $0$-motive defined by the (geometric) Lang-N\'eron group.