Galois actions on Néron models of Jacobians

Halle, Lars H.

Annales de l'Institut Fourier, Tome 60 (2010) no. 3, pp. 853-903.

Soit $X$ une courbe lisse définie sur le corps des fractions $K$ d’un anneau de valuation discrète $R$. Nous étudions une filtration naturelle sur la fibre spéciale du modèle de Néron de la Jacobienne de $X$ par des sous-schémas en groupes fermés unipotents. Nous démontrons que les sauts de cette filtration ne dépendent que du type de la fibre spéciale du modèle minimal régulier à croisements normaux stricts de $X$ sur $R$. En particulier, les sauts sont indépendants de la caractéristique résiduelle. Ensuite, nous obtenons des informations plus précises sur les sauts, et nous les calculons pour chaque type de fibre possible pour les courbes de genre 1 et 2.

Let $X$ be a smooth curve defined over the fraction field $K$ of a complete discrete valuation ring $R$. We study a natural filtration of the special fiber of the Néron model of the Jacobian of $X$ by closed, unipotent subgroup schemes. We show that the jumps in this filtration only depend on the fiber type of the special fiber of the minimal regular model with strict normal crossings for $X$ over $R$, and in particular are independent of the residue characteristic. Furthermore, we obtain information about where these jumps occur. We also compute the jumps for each of the finitely many possible fiber types for curves of genus $1$ and $2$.

Classification : 14D06
Keywords: Models of curves, tame cyclic quotient singularities, group actions on cohomology, Néron models
Mot clés : modèles des courbes, modèles de Néron, singularitś quotient cycliques modérées, actions de groupe sur la cohomologie

[1] Artin, M.; Winters, G. Degenerate fibres and stable reduction of curves, Topology, Volume 10 (1971), pp. 373-383 | DOI | MR | Zbl

[2] Bosch, Siegfried; Lütkebohmert, Werner; Raynaud, Michel Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 21, Springer-Verlag, Berlin, 1990 | MR | Zbl

[3] Chai, Ching-Li Néron models for semiabelian varieties: congruence and change of base field, Asian J. Math., Volume 4 (2000) no. 4, pp. 715-736 (Loo-Keng Hua: a great mathematician of the twentieth century) | MR | Zbl

[4] Conrad, Brian; Edixhoven, Bas; Stein, William $J_1\left(p\right)$ has connected fibers, Doc. Math., Volume 8 (2003), p. 331-408 (electronic) | MR | Zbl

[5] Deligne, P.; Mumford, D. The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. (1969) no. 36, pp. 75-109 | DOI | Numdam | MR | Zbl

[6] Deschamps, Mireille Réduction semi-stable, in Séminaire sur les Pinceaux de Courbes de Genre au Moins Deux, Astérisque, 86, Société Mathématique de France, Paris, 1981 | MR | Zbl

[7] Donovan, Peter The Lefschetz-Riemann-Roch formula, Bull. Soc. Math. France, Volume 97 (1969), pp. 257-273 | Numdam | MR | Zbl

[8] Edixhoven, Bas Néron models and tame ramification, Compositio Math., Volume 81 (1992) no. 3, pp. 291-306 | Numdam | MR | Zbl

[9] Grothendieck, Alexander Cohomologie $l$-adique et fonctions $L$, Lecture Notes in Mathematics, Vol. 589, Springer-Verlag, Berlin, 1977 Séminaire de Géometrie Algébrique du Bois-Marie 1965–1966 (SGA 5), édité par Luc Illusie | MR

[10] Halle, Lars Halvard Stable reduction of curves and tame ramification, KTH, Stockholm (2007) (Ph. D. Thesis Electronic version available at http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-4494)

[11] Hartshorne, Robin Algebraic geometry, Springer-Verlag, New York, 1977 (Graduate Texts in Mathematics, No. 52) | MR | Zbl

[12] Köck, Bernhard The Grothendieck-Riemann-Roch theorem for group scheme actions, Ann. Sci. École Norm. Sup. (4), Volume 31 (1998) no. 3, pp. 415-458 | Numdam | MR | Zbl

[13] Kodaira, K. On compact analytic surfaces. II, III, Ann. of Math. (2) 77 (1963), 563–626; ibid., Volume 78 (1963), pp. 1-40 | MR | Zbl

[14] Lipman, Joseph Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. (1969) no. 36, pp. 195-279 | DOI | Numdam | MR | Zbl

[15] Liu, Qing Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, 6, Oxford University Press, Oxford, 2002 | MR | Zbl

[16] Namikawa, Yukihiko; Ueno, Kenji The complete classification of fibres in pencils of curves of genus two, Manuscripta Math., Volume 9 (1973), pp. 143-186 | DOI | MR | Zbl

[17] Ogg, A. P. On pencils of curves of genus two, Topology, Volume 5 (1966), pp. 355-362 | DOI | MR | Zbl

[18] Serre, Jean-Pierre Linear representations of finite groups, Springer-Verlag, New York, 1977 (Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, Vol. 42) | MR

[19] Serre, Jean-Pierre Local fields, Graduate Texts in Mathematics, 67, Springer-Verlag, New York, 1979 | MR | Zbl

[20] Winters, Gayn B. On the existence of certain families of curves, Amer. J. Math., Volume 96 (1974), pp. 215-228 | DOI | MR | Zbl