Serre’s tensor construction and moduli of abelian schemes

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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Manuscripta Mathematica Springer Journals

Serre’s tensor construction and moduli of abelian schemes

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag GmbH Germany
Subject
Mathematics; Mathematics, general; Algebraic Geometry; Topological Groups, Lie Groups; Geometry; Number Theory; Calculus of Variations and Optimal Control; Optimization
ISSN
0025-2611
eISSN
1432-1785
D.O.I.
10.1007/s00229-017-0976-x
Publisher site
See Article on Publisher Site

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, 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.

Journal

Manuscripta MathematicaSpringer Journals

Published: Oct 4, 2017

References

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