Serre's Tensor Construction and Moduli of Abelian Schemes

TL;DR

This paper explores Serre's tensor construction for abelian schemes with ring actions, showing how to equip these schemes with polarizations via hermitian forms, and relates moduli spaces of CM abelian schemes to Shimura varieties.

Contribution

It establishes conditions for polarizations on Serre-constructed abelian schemes and connects moduli spaces of CM abelian schemes with integral models of Shimura varieties.

Findings

Polarizations correspond to hermitian forms on modules.

All abelian schemes in certain moduli spaces are Serre constructions.

Morphisms between objects are described via constituent data.

Abstract

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