The Drinfeld Modular Jacobian J 1 (n) has connected fibers
Annales de l'Institut Fourier, Volume 57 (2007) no. 4, pp. 1217-1252.

We study the integral model of the Drinfeld modular curve X 1 (n) for a prime n𝔽 q [T]. A function field analogue of the theory of Igusa curves is introduced to describe its reduction mod n. A result describing the universal deformation ring of a pair consisting of a supersingular Drinfeld module and a point of order n in terms of the Hasse invariant of that Drinfeld module is proved. We then apply Jung-Hirzebruch resolution for arithmetic surfaces to produce a regular model of X 1 (n) which, after contractions in the special fiber, gives a regular model with geometrically integral fiber over n. Thus the mod n component group of J 1 (n) is trivial, i.e. J 1 (n) has connected fibers.

Nous étudions le modèle intégral de la courbe modulaire X 1 (n) de Drinfeld pour un élément irreductible n𝔽 q [T]. Un analogue du corps de fonctions de la théorie des courbes d’Igusa est introduit pour décrire sa réduction mod n. Un résultat décrivant l’anneau universel de déformation d’une paire se composant d’un module de Drinfeld supersingulier et d’un point d’ordre n en termes de l’invariant de Hasse de ce module de Drinfeld est prouvé. Nous appliquons alors la résolution de Jung-Hirzebruch afin que les surfaces arithmétiques produisent un modèle régulier de X 1 (n) qui, après des contractions dans la fibre spéciale, donne un modèle régulier tel que la fibre au-dessus de n est géométriquement intègre. Ainsi, la réduction mod n du groupe des composants de J 1 (n) est triviale, c’est-à-dire les fibres de J 1 (n) sont connexes.

DOI: 10.5802/aif.2292
Classification: 11F52, 14H40, 14L05, 11G09
Keywords: Component groups, Drinfeld modular curves, Igusa curves
Mot clés : Groupes composants, courbes modulaires de Drinfeld, courbes de Igusa

Shastry, Sreekar M. 1

1 Tata Institute of Fundamental Research School of Mathematics Dr Homi Bhabha Rd Mumbai 400 005 (India)
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Shastry, Sreekar M. The Drinfeld Modular Jacobian $J_1(n)$ has connected fibers. Annales de l'Institut Fourier, Volume 57 (2007) no. 4, pp. 1217-1252. doi : 10.5802/aif.2292. https://aif.centre-mersenne.org/articles/10.5802/aif.2292/

[1] Altman, A.; Kleiman, S. Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin, 1970 | MR | Zbl

[2] Bosch, S.; Lütkebohmert, W.; Raynaud, M. Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete, 21, Springer-Verlag, Berlin, 1990 | MR | Zbl

[3] Conrad, B.; Edixhoven, B.; Stein, W. J 1 (p) has connected fibers, Doc. Math., Volume 8 (2003), p. 331-408 (electronic) | MR | Zbl

[4] Deligne, P.; Rapoport, M. Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, p. 143-316. Lecture Notes in Math., Vol. 349 | MR | Zbl

[5] Drinfeld, V. G. Elliptic modules, Mat. Sb. (N.S.), Volume 94(136) (1974), p. 594-627, 656 | MR | Zbl

[6] Fontaine, J.-M. Groupes p-divisibles sur les corps locaux, Société Mathématique de France, Paris, 1977 | MR | Zbl

[7] Freitag, E.; Kiehl, R. Étale cohomology and the Weil conjecture, Ergebnisse der Mathematik und ihrer Grenzgebiete, 13, Springer-Verlag, Berlin, 1988 | MR | Zbl

[8] Gekeler, E.-U. Zur Arithmetik von Drinfeld-Moduln, Math. Ann., Volume 262 (1983) no. 2, pp. 167-182 | DOI | MR | Zbl

[9] Gekeler, E.-U. Über Drinfeldsche Modulkurven vom Hecke-Typ, Compositio Math., Volume 57 (1986) no. 2, pp. 219-236 | Numdam | MR | Zbl

[10] Gekeler, E.-U. de Rham cohomology and the Gauss-Manin connection for Drinfeld modules, p-adic analysis (Trento, 1989) (Lecture Notes in Math.), Volume 1454, Springer, Berlin, 1990, pp. 223-255 | MR | Zbl

[11] Gekeler, E.-U. de Rham cohomology for Drinfeld modules, Séminaire de Théorie des Nombres, Paris 1988–1989 (Progr. Math.), Volume 91, Birkhäuser Boston, Boston, MA, 1990, pp. 57-85 | MR | Zbl

[12] Gekeler, E.-U. On finite Drinfeld modules, J. Algebra, Volume 141 (1991) no. 1, pp. 187-203 | DOI | MR | Zbl

[13] Goss, D. π-adic Eisenstein series for function fields, Compositio Math., Volume 41 (1980) no. 1, pp. 3-38 | Numdam | MR | Zbl

[14] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. (1965) no. 24, pp. 231 | Numdam | MR | Zbl

[15] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III., Inst. Hautes Études Sci. Publ. Math. (1966) no. 28, pp. 255 | Numdam | MR | Zbl

[16] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. IV, Inst. Hautes Études Sci. Publ. Math. (1967) no. 32, pp. 361 | Numdam | MR

[17] Grothendieck, A. Revêtements étales et groupe fondamental, Springer-Verlag, Berlin, 1971 | MR

[18] Hazewinkel, M. Formal groups and applications, Pure and Applied Mathematics, 78, Academic Press Inc., New York, 1978 | MR | Zbl

[19] Humphreys, J. E. Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, 9, Springer-Verlag, New York, 1978 | MR | Zbl

[20] Jeon, D.; Kim, C.H. On the Drinfeld modular curves X 1 (n), J. Number Theory, Volume 102 (2003) no. 2, pp. 214-222 | DOI | MR | Zbl

[21] Katz, N.; Mazur, B. Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, 108, Princeton University Press, Princeton, NJ, 1985 | MR | Zbl

[22] Laumon, G. Cohomology of Drinfeld modular varieties. Part I, Cambridge Studies in Advanced Mathematics, 41, Cambridge University Press, Cambridge, 1996 | MR | Zbl

[23] Lehmkuhl, T. Compactification of the Drinfeld Modular Surfaces (Unpublished)

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

[25] Matsumura, H. Commutative ring theory, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, Cambridge, 1989 | MR | Zbl

[26] Schlessinger, M. Functors of Artin rings, Trans. Amer. Math. Soc., Volume 130 (1968), pp. 208-222 | DOI | MR | Zbl

[27] Taguchi, Y. Semi-simplicity of the Galois representations attached to Drinfeld modules over fields of “infinite characteristics”, J. Number Theory, Volume 44 (1993) no. 3, pp. 292-314 | DOI | Zbl

[28] Tate, J. T. p-divisible groups, Proc. Conf. Local Fields (Driebergen, 1966), Springer, Berlin, 1967, pp. 158-183 | MR | Zbl

[29] Teitelbaum, J. Modular symbols for F q (T), Duke Math. J., Volume 68 (1992) no. 2, pp. 271-295 | DOI | MR | Zbl

[30] Yasufuku, Y. Deformation Theory of Formal Modules (2000) (Unpublished)

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