Dimension of the motivic Galois group of a 1-motive

TL;DR

This paper explicitly computes the dimension of the motivic Galois group for 1-motives over complex numbers, linking it to the rank of a generated multiplicative group, and offers a new perspective on the Grothendieck–André periods Conjecture.

Contribution

It provides an explicit formula for the motivic Galois group dimension of 1-motives and reformulates the periods conjecture within this context.

Findings

Explicit dimension formula for the motivic Galois group of 1-motives.

Connection between the group dimension and the rank of a multiplicative group.

New formulation of the Grothendieck–André periods Conjecture.

Abstract

We compute the dimension of the motivic Galois group of a 1-motive M defined over the field of complex numbers, expressing it explicitly in terms of the rank of the multiplicative group generated by the points defining M. As an application, we obtain a new formulation of the Grothendieck--Andr\'e periods Conjecture in the setting of 1-motives.