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The Mumford-Tate group of 1-motives

Cristiana Bertolin — 2002

Annales de l’institut Fourier

In this paper we study the structure and the degeneracies of the Mumford-Tate group $M T (M)$ of a 1-motive $M$ defined over $ℂ$ . This group is an algebraic $ℚ$ - group acting on the Hodge realization of $M$ and endowed with an increasing filtration $W_{•}$ . We prove that the unipotent radical of $M T (M)$ , which is $W_{- 1} (M T (M))$ , injects into a “generalized” Heisenberg group. We then explain how to reduce to the study of the Mumford-Tate group of a direct sum of 1-motives whose torus’character group and whose lattice are both of rank 1....

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