Mumford-Tate groups of 1-motives and Weil pairing

Abstract

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 Galmot(M). Fixing periods matrices ΠM and ΠM⁎ associated respectively to a 1-motive M and to its Cartier dual M⁎, 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.