Construction of abelian varieties with given monodromy

Abstract.

We show that certain algebraic subgroups of a unitary group that is compact at all archimedean places arise as monodromy groups of families of abelian varieties in characteristic p ≠ 0. We construct these families by deforming a $\mathbb{F}^{{ac}}_{p} - {\text{valued point}}$ on a PEL-Shimura variety that is associated to a related unitary group.