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In this paper we investigate the image of the l-adic representation attached to the Tate module of an abelian variety defined over a number field. We consider simple abelian varieties of type III in the Albert classification. We compute the image of the l-adic and mod l Galois representations and we prove the Mumford-Tate and Lang conjectures for a wide class of simple abelian varieties of type III.

On the image of Galois $l$-adic representations for abelian varieties of type III