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manuscripta mathematica

, Volume 153, Issue 3–4, pp 323–330 | Cite as

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

  • David HolmesEmail author
Open Access
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Abstract

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.

Mathematics Subject Classification

Primary 11G10 Secondary 14K30 14K05 

References

  1. 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). http://www.jstor.org/stable/j.ctt13x10qw
  2. 2.
    Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron Models. Springer, Berlin (1990)CrossRefzbMATHGoogle Scholar
  3. 3.
    Cadoret, A., Tamagawa, A.: Uniform boundedness of p-primary torsion of abelian schemes. Invent. Math. 188(1), 83–125 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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)Google Scholar
  5. 5.
    de Jong, A.J.: Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math. 83, 51–93 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Edixhoven, B.: On Néron models, divisors and modular curves. J. Ramanujan Math. Soc. 13(2), 157–194 (1998)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Holmes, D.: A Néron model of the universal jacobian. http://arxiv.org/abs/1412.2243 (2014)
  8. 8.
    Holmes, D.: Néron models of jacobians over base schemes of dimension greater than 1. J. Reine Angew. Math. http://arxiv.org/abs/1402.0647 (2014)
  9. 9.
    Holmes, D.: Torsion points and height jumping in higher-dimensional families of abelian varieties. arXiv preprint http://arxiv.org/abs/1604.04563v1 (2016)
  10. 10.
    Kambayashi, T., Miyanishi, M., Takeuchi, M.: Unipotent Algebraic Groups. Lecture Notes in Mathematics, vol. 414. Springer, Berlin (1974)CrossRefzbMATHGoogle Scholar
  11. 11.
    Silverman, J.H.: Heights and the specialization map for families of abelian varieties. J. Reine Angew. Math. 342, 197–211 (1983)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Tate, J.: Variation of the canonical height of a point depending on a parameter. Am. J. Math. 105(1), 287–294 (1983)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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.

Authors and Affiliations

  1. 1.Universiteit LeidenLeidenNetherlands

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