On the Mumford-Tate conjecture for 1-motives

TL;DR

This paper extends the Mumford-Tate conjecture to 1-motives by analyzing unipotent parts of associated groups, comparing Hodge groups, Galois images, and motivic fundamental groups, with an added adelic perspective.

Contribution

It proves the Mumford-Tate conjecture analogue for 1-motives' unipotent parts, including an adelic refinement of Galois representation images.

Findings

The unipotent part of the Hodge group matches the Galois image for 1-motives.

The results include an adelic statement about Galois representations.

Minor corrections and an appendix by P. Deligne are included.

Abstract

We show that the statement analogous to the Mumford-Tate conjecture for abelian varieties holds for 1-motives on unipotent parts. This is done by comparing the unipotent part of the associated Hodge group and the unipotent part of the image of the absolute Galois group with the unipotent part of the motivic fundamental group. Contains an appendix by P. Deligne. In this version: Minor corrections here and there. The Main result about the image of the Galois representations is now enriched by an adelic statement.