Néron models and the height jump divisor

Biesel, Owen; Holmes, David; de Jong, Robin

doi:10.1090/tran/7087

Authors: Owen Biesel, David Holmes and Robin de Jong
Journal: Trans. Amer. Math. Soc. 369 (2017), 8685-8723
MSC (2010): Primary 14H10; Secondary 11G50, 14G40, 14K15.
DOI: https://doi.org/10.1090/tran/7087
Published electronically: June 27, 2017
MathSciNet review: 3710640
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Abstract: We define an algebraic analogue, in the case of jacobians of curves, of the height jump divisor introduced recently by R. Hain. We give explicit combinatorial formulae for the height jump for families of semistable curves using labelled reduction graphs. With these techniques we prove a conjecture of Hain on the effectivity of the height jump, and also give a new proof of a theorem of Tate, Silverman and Green on the variation of heights in families of abelian varieties.

References

Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822
Gregory S. Call, Variation of local heights on an algebraic family of abelian varieties, Théorie des nombres (Quebec, PQ, 1987) de Gruyter, Berlin, 1989, pp. 72–96. MR 1024553
P. Deligne, Le déterminant de la cohomologie, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 93–177 (French). MR 902592, DOI https://doi.org/10.1090/conm/067/902592
Pierre Deligne, Le lemme de Gabber, Astérisque 127 (1985), 131–150 (French). Seminar on arithmetic bundles: the Mordell conjecture (Paris, 1983/84). MR 801921
Steven Diaz and David Harbater, Strong Bertini theorems, Trans. Amer. Math. Soc. 324 (1991), no. 1, 73–86. MR 986689, DOI https://doi.org/10.1090/S0002-9947-1991-0986689-6
Gerd Faltings and Ching-Li Chai, Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 22, Springer-Verlag, Berlin, 1990. With an appendix by David Mumford. MR 1083353
William Green, Heights in families of abelian varieties, Duke Math. J. 58 (1989), no. 3, 617–632. MR 1016438, DOI https://doi.org/10.1215/S0012-7094-89-05829-8
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 217086
Richard Hain, Biextensions and heights associated to curves of odd genus, Duke Math. J. 61 (1990), no. 3, 859–898. MR 1084463, DOI https://doi.org/10.1215/S0012-7094-90-06133-2
Richard Hain, Normal functions and the geometry of moduli spaces of curves, Handbook of moduli. Vol. I, Adv. Lect. Math. (ALM), vol. 24, Int. Press, Somerville, MA, 2013, pp. 527–578. MR 3184171

D. Holmes. Néron models of jacobians over base schemes of dimension greater than $1$. Preprint, arxiv:1402.0647.

D. Holmes, A Néron model of the universal jacobian, Preprint, arxiv:1412.2243.

David Holmes and Robin de Jong, Asymptotics of the Néron height pairing, Math. Res. Lett. 22 (2015), no. 5, 1337–1371. MR 3488379, DOI https://doi.org/10.4310/MRL.2015.v22.n5.a5

James Jeans, The mathematical theory of electricity and magnetism, Cambridge University Press, New York, 1960. 5th ed. MR 0115577

A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. MR 1423020

Gérard Laumon and Laurent Moret-Bailly, 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-Verlag, Berlin, 2000 (French). MR 1771927

Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. MR 715605

Dale Allen Lear, Extensions of normal functions and asymptotics of the height pairing, ProQuest LLC, Ann Arbor, MI, 1990. Thesis (Ph.D.)–University of Washington. MR 2685578

Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné; Oxford Science Publications. MR 1917232

H. M. Melvin, On concavity of resistance functions, J. Appl. Phys. 27 (1956), 658–659.

Laurent Moret-Bailly, Pinceaux de variétés abéliennes, Astérisque 129 (1985), 266 (French, with English summary). MR 797982

Laurent Moret-Bailly, Métriques permises, Astérisque 127 (1985), 29–87 (French). Seminar on arithmetic bundles: the Mordell conjecture (Paris, 1983/84). MR 801918

André Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Inst. Hautes Études Sci. Publ. Math. 21 (1964), 128 (French). MR 179172, DOI https://doi.org/10.1007/bf02684271

Gregory Pearlstein, ${\rm SL}_2$-orbits and degenerations of mixed Hodge structure, J. Differential Geom. 74 (2006), no. 1, 1–67. MR 2260287

Michel Raynaud, Modèles de Néron, C. R. Acad. Sci. Paris Sér. A-B 262 (1966), A345–A347 (French). MR 194421

C. Shannon and D. Hagelbarger, Concavity of resistance functions, J. Appl. Phys. 27 (1956), 42–43.

Joseph H. Silverman, Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math. 342 (1983), 197–211. MR 703488, DOI https://doi.org/10.1515/crll.1983.342.197

J. Tate, Variation of the canonical height of a point depending on a parameter, Amer. J. Math. 105 (1983), no. 1, 287–294. MR 692114, DOI https://doi.org/10.2307/2374389

Shouwu Zhang, Admissible pairing on a curve, Invent. Math. 112 (1993), no. 1, 171–193. MR 1207481, DOI https://doi.org/10.1007/BF01232429