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

TL;DR

This paper classifies the possible Hodge groups of certain simple polarizable rational Hodge structures with specific Hodge numbers, extending previous results and confirming the Hodge conjecture for related abelian varieties.

Contribution

It generalizes earlier classifications of Hodge groups for structures with Hodge numbers (n,0,...,0,n), especially for n=1, 4, prime p, and n=2p, under certain conditions.

Findings

Classified Hodge groups for n=1, 4, prime p

Determined Hodge groups for n=2p with conditions on endomorphism algebra

Confirmed Hodge and General Hodge Conjectures for related abelian varieties

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.