# zbMATH — the first resource for mathematics

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.
##### MSC:
 11F52 Modular forms associated to Drinfel’d modules 14L05 Formal groups, $$p$$-divisible groups 11G09 Drinfel’d modules; higher-dimensional motives, etc.
##### Keywords:
component groups; Drinfeld modular curves; Igusa curves
Full Text:
##### References:
 [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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.