manuscripta mathematica

, Volume 156, Issue 3–4, pp 409–456 | Cite as

Serre’s tensor construction and moduli of abelian schemes

  • Zavosh Amir-Khosravi


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.

Mathematics Subject Classification

11G15 14K10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Braun, H., Koecher, M.: Jordan–Algebren, Die Grundlehren Der Mathematischen Wissenschaften In Einzeldarstellungen, vol. 128. Springer, Berlin (1966)Google Scholar
  2. 2.
    Chai, C.-L., Conrad, B., Oort, F.: Complex Multiplication and Lifting Problems. Mathematical Surveys and Monographs, vol. 195. American Mathematical Society, Providence (2014)MATHGoogle Scholar
  3. 3.
    Conner, P.E., Hurrelbrink, J.: Class Number Parity. Series in Pure Mathematics, vol. 8. World Scientific Publishing Co., Singapore (1988)CrossRefMATHGoogle Scholar
  4. 4.
    Conrad, B.: Gross–Zagier revisited. In: Heegner Points and Rankin \(L\)-Series. Math. Sci. Res. Inst. Publ., vol. 49, pp. 67–163. Cambridge University Press, Cambridge (2004). With an appendix by W. R. MannGoogle Scholar
  5. 5.
    Deligne, P.: Catégories tannakiennes. In: The Grothendieck Festschrift, vol. II. Progr. Math., vol. 87, pp. 111–195. Birkhäuser Boston, Boston (1990)Google Scholar
  6. 6.
    Faltings, G., Chai, C.-L.: Degeneration of abelian varieties. In: Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 22. Springer, Berlin (1990). With an appendix by David MumfordGoogle Scholar
  7. 7.
    Grayson, D.: Higher algebraic \(K\)-theory. II (after Daniel Quillen). In: Algebraic \(K\)-Theory (Proc. Conf., Northwestern Univ., Evanston, 1976). Lecture Notes in Math., vol. 551, pp. 217–240. Springer, Berlin (1976)Google Scholar
  8. 8.
    Grothendieck, A., Demazure, M.: Schémas en groupes (SGA 3). Tome I. Propriétés générales des schémas en groupes. Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 7. Société Mathématique de France, Paris, 2011. Séminaire de Géométrie Algébrique du Bois Marie 1962–64. [Algebraic Geometry Seminar of Bois Marie 1962–64], A seminar directed by M. Demazure and A. Grothendieck with the collaboration of M. Artin, J.-E. Bertin, P. Gabriel, M. Raynaud and J-P. Serre, Revised and annotated edition of the 1970 French originalGoogle Scholar
  9. 9.
    Hida, H.: \(p\)-Adic Automorphic Forms on Shimura Varieties. Springer Monographs in Mathematics. Springer, New York (2004)CrossRefMATHGoogle Scholar
  10. 10.
    Howard, B.: Complex multiplication cycles and Kudla–Rapoport divisors. Ann. Math. (2) 176(2), 1097–1171 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Howard, B.: Complex multiplication cycles and Kudla–Rapoport divisors, II. Am. J. Math. 137(3), 639–698 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kudla, S., Rapoport, M.: Special cycles on unitary Shimura varieties I. Unramified local theory. Invent. Math. 184(3), 629–682 (2011)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kudla, S., Rapoport, M.: Special cycles on unitary Shimura varieties II: global theory. J. Reine Angew. Math. 697, 91–157 (2014)MathSciNetMATHGoogle Scholar
  14. 14.
    Lan, K.-W.: Arithmetic Compactifications of PEL-Type Shimura Varieties. London Mathematical Society Monographs Series, vol. 36. Princeton University Press, Princeton (2013)Google Scholar
  15. 15.
    Laumon, G., Moret-Bailly, L.: Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 39. Springer, Berlin (2000)Google Scholar
  16. 16.
    Lauter, K.: The maximum or minimum number of rational points on genus three curves over finite fields. Compos. Math. 134(1), 87–111 (2002). With an appendix by Jean-Pierre SerreGoogle Scholar
  17. 17.
    Lane, S.M.: Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5, 2nd edn. Springer, New York (1998)Google Scholar
  18. 18.
    McCrimmon, K.: A Taste of Jordan Algebras. Universitext. Springer, New York (2004)MATHGoogle Scholar
  19. 19.
    Milne, J.S.: Étale Cohomology. Princeton Mathematical Series, vol. 33. Princeton University Press, Princeton (1980)Google Scholar
  20. 20.
    Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, 3rd edn. Springer, Berlin (1994)Google Scholar
  21. 21.
    Mumford, D.: Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5. Tata Institute of Fundamental Research, Bombay (2008). With appendices by C. P. Ramanujam and Yuri Manin, Corrected reprint of the second edition (1974)Google Scholar
  22. 22.
    Pappas, G.: On the arithmetic moduli schemes of PEL Shimura varieties. J. Algebraic Geom. 9(3), 577–605 (2000)MathSciNetMATHGoogle Scholar
  23. 23.
    Rapoport, M.: Compactifications de l’espace de modules de Hilbert–Blumenthal. Compos. Math. 36(3), 255–335 (1978)MathSciNetMATHGoogle Scholar
  24. 24.
    Rivano, N.S.: Catégories Tannakiennes. In: Lecture Notes in Mathematics, vol. 265. Springer, Berlin (1972)Google Scholar
  25. 25.
    Serre, J.-P., Tate, J.: Good reduction of abelian varieties. Ann. Math. 2(88), 492–517 (1968)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Shimura, G.: Abelian Varieties with Complex Multiplication and Modular Functions. Princeton Mathematical Series, vol. 46. Princeton University Press, Princeton (1998)CrossRefMATHGoogle Scholar
  27. 27.
    Tambara, D.: A duality for modules over monoidal categories of representations of semisimple Hopf algebras. J. Algebra 241(2), 515–547 (2001)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Weibel, C.A.: The \(K\)-Book, Graduate Studies in Mathematics. An Introduction to Algebraic \(K\)-Theory, vol. 145. American Mathematical Society, Providence (2013)MATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsCalifornia Institute of TechnologyPasadenaUSA

Personalised recommendations