Hodge Groups of Hodge Structures with Hodge Numbers

This paper studies the possible Hodge groups of simple polarizable Q-Hodge structures with Hodge numbers (n, 0, . . . , 0, n). In particular, it generalizes earlier work of Ribet and MoonenZarhin to completely determine the possible Hodge groups of such Hodge structures when n is equal to 1, 4, or a prime p. In addition, the paper determines possible Hodge groups, under certain conditions on the endomorphism algebra, when n = 2p, for p an odd prime. A consequence of these results is that both the Hodge and General Hodge Conjectures hold for all powers of a simple 2p-dimensional abelian variety satisfying the aforementioned conditions.