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.
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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
Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron Models. Springer, Berlin (1990)
Cadoret, A., Tamagawa, A.: Uniform boundedness of p-primary torsion of abelian schemes. Invent. Math. 188(1), 83–125 (2012)
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)
de Jong, A.J.: Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math. 83, 51–93 (1996)
Edixhoven, B.: On Néron models, divisors and modular curves. J. Ramanujan Math. Soc. 13(2), 157–194 (1998)
Holmes, D.: A Néron model of the universal jacobian. http://arxiv.org/abs/1412.2243 (2014)
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)
Holmes, D.: Torsion points and height jumping in higher-dimensional families of abelian varieties. arXiv preprint http://arxiv.org/abs/1604.04563v1 (2016)
Kambayashi, T., Miyanishi, M., Takeuchi, M.: Unipotent Algebraic Groups. Lecture Notes in Mathematics, vol. 414. Springer, Berlin (1974)
Silverman, J.H.: Heights and the specialization map for families of abelian varieties. J. Reine Angew. Math. 342, 197–211 (1983)
Tate, J.: Variation of the canonical height of a point depending on a parameter. Am. J. Math. 105(1), 287–294 (1983)
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Holmes, D. Quasi-compactness of Néron models, and an application to torsion points. manuscripta math. 153, 323–330 (2017). https://doi.org/10.1007/s00229-016-0887-2
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DOI: https://doi.org/10.1007/s00229-016-0887-2