Serre's tensor construction and moduli of abelian schemes
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, étale 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.
© Springer-Verlag GmbH Germany 2017. Received: 15 August 2015 / Accepted: 19 September 2017. Published online: 4 October 2017. Much of the content in this article is part of the author's thesis, written at the University of Toronto under the supervision of Stephen S. Kudla. I thank Professor Kudla for suggesting the problem and for subsequent encouragements. I thank Brian Conrad and Florian Herzig for looking over an early draft, and offering helpful comments and suggestions. I also thank Ben Howard, who read the first version carefully, spottingmistakes and suggesting improvements. In particular the ideal condition J_Φ Lie_S(A) = 0 in the definition of M^n_Φ was suggested by him. I express gratitude to the anonymous referee for patiently reading the article, uncovering many mistakes, and suggesting significant improvements and simplifications, especially in the proofs of the final sections.
Accepted Version - 1507.07607