Skip to main content

Quasi-compactness of Néron models, and an application to torsion points


We prove that Néron models of Jacobians of generically-smooth nodal curves over bases of arbitrary dimension are quasi-compact (hence of finite type) whenever they exist. We give a simple application to the orders of torsion subgroups of Jacobians over number fields.


  1. Artin, M.: Algebraization of Formal Moduli. I. In: Spencer, D.C., Iyanaga S. (eds.) Global Analysis: Papers in Honor of K. Kodaira (PMS-29). Princeton University Press, pp. 21–71 (1969).

  2. Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron Models. Springer, Berlin (1990)

    Book  MATH  Google Scholar 

  3. Cadoret, A., Tamagawa, A.: Uniform boundedness of p-primary torsion of abelian schemes. Invent. Math. 188(1), 83–125 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  4. Cadoret, A., Tamagawa, A.: Note on torsion conjecture. In: Geometric and Differential Galois Theories, Volume 27 of Séminar Congress, pp. 57–68. Society Mathematics France, Paris (2013)

  5. de Jong, A.J.: Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math. 83, 51–93 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  6. Edixhoven, B.: On Néron models, divisors and modular curves. J. Ramanujan Math. Soc. 13(2), 157–194 (1998)

    MathSciNet  MATH  Google Scholar 

  7. Holmes, D.: A Néron model of the universal jacobian. (2014)

  8. Holmes, D.: Néron models of jacobians over base schemes of dimension greater than 1. J. Reine Angew. Math. (2014)

  9. Holmes, D.: Torsion points and height jumping in higher-dimensional families of abelian varieties. arXiv preprint (2016)

  10. Kambayashi, T., Miyanishi, M., Takeuchi, M.: Unipotent Algebraic Groups. Lecture Notes in Mathematics, vol. 414. Springer, Berlin (1974)

    Book  MATH  Google Scholar 

  11. Silverman, J.H.: Heights and the specialization map for families of abelian varieties. J. Reine Angew. Math. 342, 197–211 (1983)

    MathSciNet  MATH  Google Scholar 

  12. Tate, J.: Variation of the canonical height of a point depending on a parameter. Am. J. Math. 105(1), 287–294 (1983)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to David Holmes.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Holmes, D. Quasi-compactness of Néron models, and an application to torsion points. manuscripta math. 153, 323–330 (2017).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

Mathematics Subject Classification

  • Primary 11G10
  • Secondary 14K30
  • 14K05