Moduli of Abelian Varieties

  • Ching-Li Chai
Conference paper


This is the written version of my talk given on Oct. 11, 1990. Its very limited purpose is to outline a picture about the moduli of abelian varieties. Only a small portion of the theory of moduli of abelian varieties is covered. Many (if not all) statements are imprecise and no proof is offered, in the hope that this makes the material more palatable. For the grumbling readers, the precise statements together with proofs can be found in the references.


Modulus Space Abelian Variety Discrete Valuation Ring Invertible Sheaf Siegel Modular Form 
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  1. [A 1]
    M. Artin, Algebraic approximation of structures over complete local rings, Publ. Math. I.H.E.S. no. 36 (1969), 23–58.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [A 2]
    M. Artin, Algebraization of formal moduli I, Global Analysis, Papers in Honor of K. Kodaira, Princeton Univ. Press, Princeton, 1969, 21–71.Google Scholar
  3. [A 3]
    M. Artin, Algebraization of formal moduli II, Ann. of Math. 91 (1970), 88–135.MathSciNetCrossRefGoogle Scholar
  4. [A 4]
    M. Artin, Versal deformations and algebraic stacks, In. Math. 27 (1974), 165–89.MathSciNetzbMATHGoogle Scholar
  5. [AMRT]
    A. Ash, D. Mumford, M. Rapoport, Y.-S. Tai, Smooth Compactification of Locally Symmetric Varieties, Math. Sci. Press, 1975.zbMATHGoogle Scholar
  6. [B]
    L. Breen, Fonctions Thêta et Théorème du cube, (Lecture Notes in Math. 980), Springer-Verlag, 1983.zbMATHGoogle Scholar
  7. [C 1]
    C.-L. Chai, Compactification of Siegel Moduli Schemes, (London Math. Soc. Lecture Notes Series, no. 107), Cambridge Univ. Press, 1985.CrossRefzbMATHGoogle Scholar
  8. [C 2]
    C.-L. Chai, Arithmetic compactifiaction of the Siegel moduli space, Proc. Symp. Pure Math. 49, (Proceedings of the 1987 AMS Summer School on Theta Functions) 47, Amer. Math. Soc, Part. 2, 1989, 19–44.CrossRefGoogle Scholar
  9. [C 3]
    C.-L. Chai, Singularities of the Γ0(p)-level structure I, preprint, 1990.Google Scholar
  10. [CN 1]
    C.-L. Chai and P. Norman, Bad reduction of the Siegel moduli schemes of genus two with Γ0(p)-level structure, to appear in Amer. J. Math. Google Scholar
  11. [CN 2]
    C.-L. Chai and P. Norman, Singularities of the Γ0(p)-level structure II, preprint, 1990.Google Scholar
  12. [DEP]
    C. Deconcini, E. Eisenbud and C. Procesi, Hodge Algebras, (Astérisque 91), 1982.Google Scholar
  13. [[DR]]
    P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular Functions in One Variable II, (Lecture Notes in Mathematics 349), Springer-Verlag, 1973, 143–316.Google Scholar
  14. [DRS]
    P. Doubilet, G.-C. Rota and J. Stein, On the foundations of combinatorial theory IX: Combinatorial methods in invariant theory, Studies in Appl. Math. 53 (1974), 185–216.MathSciNetzbMATHGoogle Scholar
  15. [DM]
    P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Pubi. Math. I.H.E.S. 36 (1969), 75–109.MathSciNetzbMATHGoogle Scholar
  16. [F 1]
    G. Faltings, Endlichkeilssätze für abelsche Varietäten über Zahlkörpern, Inv. Math. 73 (1983), 349–366.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [F 2]
    G. Faltings, Arithmetische Kompatifizierung des Modulraums der abelschen Varietäten, Lecture Note in Math. no. 1111, Springer-Verlag, 1985, 321–383.Google Scholar
  18. [F 3]
    G. Faltings, Crystalline cohomology and p-adic Galois-representations, Proc. of the JAMI Inaugural Conf., ed. by J.-I. Igusa, supp. to the Amer. J. Math., 1989, 25–80.Google Scholar
  19. [F 5]
    G. Faltings, F-isocrystals on open varieties: Results and conjectures, Grothendieck Festschrift, Progress in Math., Birkhäuser, 1990.Google Scholar
  20. [FC]
    G. Faltings and C.-L. Chai, Degeneration of Abelian Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 22, Springer-Verlag, 1990.Google Scholar
  21. [I 1]
    J.-I. Igusa, On the theory of compactifications, AMS Summer Institute on Algebraic Geometry, Woods Hole 1964, mimeographed lecture notes.Google Scholar
  22. [I 2]
    J.-I. Igusa, A desingularization problem in the theory of Siegel modular functions, Math. Ann. 168 (1967), 228–260.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [Ka]
    K. Kato, Logarithmic structures of Fontaine-Illusie, Proc. of the JAMI Inaugural Conf., ed. by J.-I. Igusa, supp. to the Amer. J. Math., 1989, 191–224.Google Scholar
  24. [Ku]
    V. Kulikov, Degenerations of K3 surfaces and Enriques surfaces, Math. USSR Izvestija refbold 11 (1977), 957–989.CrossRefzbMATHGoogle Scholar
  25. [KM]
    N. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, (Annals of Math. Studies, no. 108), Princeton Univ. Press, 1985.zbMATHGoogle Scholar
  26. [MB]
    L. Moret-Bailly, Pinceaux de Variétés Abéliennes, (Astérisque 129), 1985.zbMATHGoogle Scholar
  27. [Na 1]
    Y. Namikawa, A new compactification of the Siegel space and degeneration of abelian varieties I, II, Math. Ann. 221 (1976), 97–141; ibid., 201–241.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [Nu 2]
    Y. Namikawa, Toroidal degenerationof abelian varieties II, Math. Ann. 245 (1979), 117–150.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [No]
    P. Norman, An algorithm for computing local moduli of abelian varieties, Ann. of Math. 101, (1975), 317–334.MathSciNetCrossRefGoogle Scholar
  30. [NO]
    P. Norman and F. Oort, Moduli of abelian varieties, Ann. of Math. 112 (1980), 413–439.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [GIT]
    D. Mumford and J. Forgarty, Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 2. Folge, Band 24, Springer-Verlag, 1982, 2nd ed.Google Scholar
  32. [AV]
    D. Mumford, Abelian Varieties, Tata Inst. Fund. Research Studies in Math. vol. 5, Oxford Univ. Press, 1970.Google Scholar
  33. [M]
    D. Mumford, An analytic construction of degenerating abelian varieties over complete rings, Compositio Math., 24 (1972), 239–272.MathSciNetzbMATHGoogle Scholar
  34. [SGA 1]
    A. Grothendieck, Revêtements Etales et Groupe Fondamental, (Lecture Notes in Math. no. 224), Springer-Verlag, 1971.zbMATHGoogle Scholar
  35. [SGA 7I]
    A. Grothendieck, Groupes de Monodromie en Géométrie Algébrique I, (Lecture Notes in Math. no. 288) , Springer-Verlag, 1972.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Ching-Li Chai
    • 1
  1. 1.University of PennsylvaniaPhiladelphiaUSA

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