Dimension of the motivic Galois group of a 1-motive

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--André periods Conjecture in the setting of 1-motives.

The treatment of the quotient torus $Z(1)/Z^{\prime }(1)$ has been substantially revised. In particular, we show that LieBracket fibres contribute not only to the torus $Z^{\prime }(1)$ but also to the quotient torus $Z(1)/Z^{\prime }(1).$