Mumford-Tate groups of 1-motives and Weil pairing

We show how the geometry of a 1 -motive M (that is existence of endomorphisms and relations between the points defining it) determines the dimension of its motivic Galois group G al mot ( M ). Fixing periods matrices Pi M and Pi M & lowast; associated respectively to a 1 -motive M and to its Cartier dual M & lowast; , we describe the action of the Mumford-Tate group of M on these matrices. In the semi-elliptic case, according to the geometry of M we classify polynomial relations between the periods of M and we compute exhaustively the matrices representing the Mumford-Tate group of M . This representation brings new light on Grothendieck periods conjecture in the case of 1 -motives. (c) 2024 Elsevier B.V. All rights reserved.