Hodge Groups of Hodge Structures with Hodge Numbers $(n,0,\ldots,0,n)$

Abstract

This paper studies the possible Hodge groups of simple polarizable $\mathbb{Q}$-Hodge structures with Hodge numbers $(n,0,\ldots,0,n)$. In particular, it generalizes earlier work of Ribet and Moonen-Zarhin 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.