The Drinfeld Modular Jacobian J 1 ( n ) has connected fibers

Sreekar M. Shastry[1]

  • [1] Tata Institute of Fundamental Research School of Mathematics Dr Homi Bhabha Rd Mumbai 400 005 (India)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 4, page 1217-1252
  • ISSN: 0373-0956

Abstract

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We study the integral model of the Drinfeld modular curve X 1 ( n ) for a prime n ∈ 𝔽 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.

How to cite

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Shastry, Sreekar M.. "The Drinfeld Modular Jacobian $J_1(n)$ has connected fibers." Annales de l’institut Fourier 57.4 (2007): 1217-1252. <http://eudml.org/doc/10256>.

@article{Shastry2007,
abstract = {We study the integral model of the Drinfeld modular curve $X_1(n)$ for a prime $n\in \mathbb\{F\}_q[T]$. 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.},
affiliation = {Tata Institute of Fundamental Research School of Mathematics Dr Homi Bhabha Rd Mumbai 400 005 (India)},
author = {Shastry, Sreekar M.},
journal = {Annales de l’institut Fourier},
keywords = {Component groups; Drinfeld modular curves; Igusa curves; component groups},
language = {eng},
number = {4},
pages = {1217-1252},
publisher = {Association des Annales de l’institut Fourier},
title = {The Drinfeld Modular Jacobian $J_1(n)$ has connected fibers},
url = {http://eudml.org/doc/10256},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Shastry, Sreekar M.
TI - The Drinfeld Modular Jacobian $J_1(n)$ has connected fibers
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 4
SP - 1217
EP - 1252
AB - We study the integral model of the Drinfeld modular curve $X_1(n)$ for a prime $n\in \mathbb{F}_q[T]$. 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.
LA - eng
KW - Component groups; Drinfeld modular curves; Igusa curves; component groups
UR - http://eudml.org/doc/10256
ER -

References

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  1. A. Altman, S. Kleiman, Introduction to Grothendieck duality theory, (1970), Springer-Verlag, Berlin Zbl0215.37201MR274461
  2. S. Bosch, W. Lütkebohmert, M. Raynaud, Néron models, 21 (1990), Springer-Verlag, Berlin Zbl0705.14001MR1045822
  3. B. Conrad, B. Edixhoven, W. Stein, J 1 ( p ) has connected fibers, Doc. Math. 8 (2003), 331-408 (electronic) Zbl1101.14311MR2029169
  4. P. Deligne, M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) (1973), 143-316. Lecture Notes in Math., Vol. 349, Springer, Berlin Zbl0281.14010MR330050
  5. V. G. Drinfeld, Elliptic modules, Mat. Sb. (N.S.) 94(136) (1974), 594-627, 656 Zbl0321.14014MR384707
  6. J.-M. Fontaine, Groupes p -divisibles sur les corps locaux, (1977), Société Mathématique de France, Paris Zbl0377.14009MR498610
  7. E. Freitag, R. Kiehl, Étale cohomology and the Weil conjecture, 13 (1988), Springer-Verlag, Berlin Zbl0643.14012MR926276
  8. E.-U. Gekeler, Zur Arithmetik von Drinfeld-Moduln, Math. Ann. 262 (1983), 167-182 Zbl0536.14028MR690193
  9. E.-U. Gekeler, Über Drinfeldsche Modulkurven vom Hecke-Typ, Compositio Math. 57 (1986), 219-236 Zbl0599.14032MR827352
  10. E.-U. Gekeler, de Rham cohomology and the Gauss-Manin connection for Drinfeld modules, -adic analysis (Trento, 1989) 1454 (1990), 223-255, Springer, Berlin Zbl0735.14016MR1094856
  11. E.-U. Gekeler, de Rham cohomology for Drinfeld modules, Séminaire de Théorie des Nombres, Paris 1988–1989 91 (1990), 57-85, Birkhäuser Boston, Boston, MA Zbl0728.14024MR1104700
  12. E.-U. Gekeler, On finite Drinfeld modules, J. Algebra 141 (1991), 187-203 Zbl0731.11034MR1118323
  13. D. Goss, π -adic Eisenstein series for function fields, Compositio Math. 41 (1980), 3-38 Zbl0422.10020MR578049
  14. A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. (1965) Zbl0135.39701MR199181
  15. A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III., Inst. Hautes Études Sci. Publ. Math. (1966) Zbl0144.19904MR217086
  16. A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. IV, Inst. Hautes Études Sci. Publ. Math. (1967) Zbl0153.22301MR238860
  17. A. Grothendieck, Revêtements étales et groupe fondamental, (1971), Springer-Verlag, Berlin MR354651
  18. M. Hazewinkel, Formal groups and applications, 78 (1978), Academic Press Inc., New York Zbl0454.14020MR506881
  19. J. E. Humphreys, Introduction to Lie algebras and representation theory, 9 (1978), Springer-Verlag, New York Zbl0254.17004MR499562
  20. D. Jeon, C.H. Kim, On the Drinfeld modular curves X 1 ( n ) , J. Number Theory 102 (2003), 214-222 Zbl1052.11041MR1997789
  21. N. Katz, B. Mazur, Arithmetic moduli of elliptic curves, 108 (1985), Princeton University Press, Princeton, NJ Zbl0576.14026MR772569
  22. G. Laumon, Cohomology of Drinfeld modular varieties. Part I, 41 (1996), Cambridge University Press, Cambridge Zbl0837.14018MR1381898
  23. T. Lehmkuhl, Compactification of the Drinfeld Modular Surfaces Zbl1179.11014
  24. Q. Liu, Algebraic geometry and arithmetic curves, 6 (2002), Oxford University Press, Oxford Zbl0996.14005MR1917232
  25. H. Matsumura, Commutative ring theory, 8 (1989), Cambridge University Press, Cambridge Zbl0666.13002MR1011461
  26. M. Schlessinger, Functors of Artin rings, Trans. Amer. Math. Soc. 130 (1968), 208-222 Zbl0167.49503MR217093
  27. Y. Taguchi, Semi-simplicity of the Galois representations attached to Drinfeld modules over fields of “infinite characteristics”, J. Number Theory 44 (1993), 292-314 Zbl0781.11024
  28. J. T. Tate, p -divisible groups, Proc. Conf. Local Fields (Driebergen, 1966) (1967), 158-183, Springer, Berlin Zbl0157.27601MR231827
  29. J. Teitelbaum, Modular symbols for F q ( T ) , Duke Math. J. 68 (1992), 271-295 Zbl0777.11021MR1191561
  30. Y. Yasufuku, Deformation Theory of Formal Modules, (2000) 

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