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Given a polarized abelian scheme with action by a ring, and a projectivefinitely presented module over that ring, Serre's tensor construction producesa new abelian scheme. We show that to equip these abelian schemes withpolarizations it's enough to equip the projective modules with non-degeneratepositive-definite hermitian forms. As an application, we relate certain modulispaces of principally polarized abelian schemes with action by the ring ofintegers of a CM field. More specifically, we consider integral models ofzero-dimensional Shimura varieties associated to compact unitary groups. Weshow that all abelian schemes in such moduli spaces are, \'etale locally overtheir base schemes, Serre constructions of CM abelian schemes withpositive-definite hermitian modules. We also describe the morphisms betweensuch objects in terms of morphisms between the constituent data, and formulatethese relations as an isomorphism of algebraic stacks.

Serre's Tensor Construction and Moduli of Abelian Schemes