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The Drinfeld modular Jacobian \(J_1 (N)\) has connected fibers. (English) Zbl 1165.11047
Let \(n\in A:= {\mathbb F}_q[T]\) be a prime element. Let \(X_1(n)\) be the smooth projective curve over \({\mathbb F}_q(T)\) associated to the moduli problem of classifying pairs \((\phi,P)\), where \(\phi\) is a rank 2 Drinfeld \(A\)-module over an \({\mathbb F}_q(T)\)-scheme and \(P\) is a nowhere vanishing point of its \(n\)-torsion.
Let \(J_1(n)\) be the Jacobian of \(X_1(n)\). The main result of this paper is that the closed fibre of the Néron model of \(J_1(n)\) over \(A_{(n)}\) has trivial geometric component group. It follows that the Néron model of \(J_1(n)\) over \({\mathbb P}^1_{{\mathbb F}_q} - \infty\) has connected fibres, hence the title. (The component group of the fibre above \(\infty\) is a much more complicated beast, but is not part of the moduli problem). This work is a function field analogue of results of B. Conrad, B. Edixhoven and W. Stein [Doc. Math., J. DMV 8, 331–408 (2003; Zbl 1101.14311)], and the proof strategy is similar, although the translation is far from trivial.
To prove the result, it suffices to show that \(X_1(n)\) has a regular proper model over \(A_{(n)}\) with geometrically integral special fibre. The author does this by first constructing a model with a unique non-regular point (corresponding to \((\phi,P)\), where \(j(\phi)=0\) and \(P\) is the kernel of Frobenius). This is a cyclic quotient singularity, which is resolved using the Jung-Hirzebruch resolution developed in [loc. cit.]. Contraction of the special fibre of this resolution then yields the desired integral model of \(X_1(n)\).
A result of independent interest obtained along the way is a function field analogue of the theory of Igusa curves. The paper is rather technical, but very well written.
11F52 Modular forms associated to Drinfel’d modules
14L05 Formal groups, \(p\)-divisible groups
11G09 Drinfel’d modules; higher-dimensional motives, etc.
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[1] Altman, A.; Kleiman, S., Introduction to Grothendieck duality theory, (1970), Springer-Verlag, Berlin · Zbl 0215.37201
[2] Bosch, S.; Lütkebohmert, W.; Raynaud, M., Néron models, 21, (1990), Springer-Verlag, Berlin · Zbl 0705.14001
[3] Conrad, B.; Edixhoven, B.; Stein, W.\(, J_1(p)\) has connected fibers, Doc. Math., 8, 331-408 (electronic), (2003) · Zbl 1101.14311
[4] Deligne, P.; Rapoport, M., Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), LES schémas de modules de courbes elliptiques, 143-316. Lecture Notes in Math., Vol. 349, (1973), Springer, Berlin · Zbl 0281.14010
[5] Drinfeld, V. G., Elliptic modules, Mat. Sb. (N.S.), 94(136), 594-627, 656, (1974) · Zbl 0321.14014
[6] Fontaine, J.-M., Groupes \(p\)-divisibles sur les corps locaux, (1977), Société Mathématique de France, Paris · Zbl 0377.14009
[7] Freitag, E.; Kiehl, R., Étale cohomology and the Weil conjecture, 13, (1988), Springer-Verlag, Berlin · Zbl 0643.14012
[8] Gekeler, E.-U., Zur arithmetik von Drinfeld-moduln, Math. Ann., 262, 2, 167-182, (1983) · Zbl 0536.14028
[9] Gekeler, E.-U., Über drinfeldsche modulkurven vom Hecke-typ, Compositio Math., 57, 2, 219-236, (1986) · Zbl 0599.14032
[10] Gekeler, E.-U.\(, p\)-adic analysis (Trento, 1989), 1454, De Rham cohomology and the Gauss-Manin connection for Drinfeld modules, 223-255, (1990), Springer, Berlin · Zbl 0735.14016
[11] Gekeler, E.-U., Séminaire de Théorie des Nombres, Paris 1988-1989, 91, De Rham cohomology for Drinfeld modules, 57-85, (1990), Birkhäuser Boston, Boston, MA · Zbl 0728.14024
[12] Gekeler, E.-U., On finite Drinfeld modules, J. Algebra, 141, 1, 187-203, (1991) · Zbl 0731.11034
[13] Goss, D.\(, π \)-adic Eisenstein series for function fields, Compositio Math., 41, 1, 3-38, (1980) · Zbl 0422.10020
[14] Grothendieck, A., Éléments de géométrie algébrique. IV. étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math., 24, 231 pp., (1965) · Zbl 0135.39701
[15] Grothendieck, A., Éléments de géométrie algébrique. IV. étude locale des schémas et des morphismes de schémas. III., Inst. Hautes Études Sci. Publ. Math., 28, 255 pp., (1966) · Zbl 0144.19904
[16] Grothendieck, A., Éléments de géométrie algébrique. IV. étude locale des schémas et des morphismes de schémas. IV, Inst. Hautes Études Sci. Publ. Math., 32, 361 pp., (1967) · Zbl 0153.22301
[17] Grothendieck, A., Revêtements étales et groupe fondamental, (1971), Springer-Verlag, Berlin
[18] Hazewinkel, M., Formal groups and applications, 78, (1978), Academic Press Inc., New York · Zbl 0454.14020
[19] Humphreys, J. E., Introduction to Lie algebras and representation theory, 9, (1978), Springer-Verlag, New York · Zbl 0254.17004
[20] Jeon, D.; Kim, C. H., On the Drinfeld modular curves \(X_1(n),\) J. Number Theory, 102, 2, 214-222, (2003) · Zbl 1052.11041
[21] Katz, N.; Mazur, B., Arithmetic moduli of elliptic curves, 108, (1985), Princeton University Press, Princeton, NJ · Zbl 0576.14026
[22] Laumon, G., Cohomology of Drinfeld modular varieties. Part I, 41, (1996), Cambridge University Press, Cambridge · Zbl 0837.14018
[23] Lehmkuhl, T., Compactification of the Drinfeld Modular Surfaces · Zbl 1179.11014
[24] Liu, Q., Algebraic geometry and arithmetic curves, 6, (2002), Oxford University Press, Oxford · Zbl 0996.14005
[25] Matsumura, H., Commutative ring theory, 8, (1989), Cambridge University Press, Cambridge · Zbl 0666.13002
[26] Schlessinger, M., Functors of Artin rings, Trans. Amer. Math. Soc., 130, 208-222, (1968) · Zbl 0167.49503
[27] Taguchi, Y., Semi-simplicity of the Galois representations attached to Drinfeld modules over fields of “infinite characteristics”, J. Number Theory, 44, 3, 292-314, (1993) · Zbl 0781.11024
[28] Tate, J. T., Proc. Conf. Local Fields (Driebergen, 1966)\(, p\)-divisible groups, 158-183, (1967), Springer, Berlin · Zbl 0157.27601
[29] Teitelbaum, J., Modular symbols for \(\textbf{F}_q(T),\) Duke Math. J., 68, 2, 271-295, (1992) · Zbl 0777.11021
[30] Yasufuku, Y., Deformation Theory of Formal Modules, (2000)
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