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On component groups of $J_0(N)$ and degeneracy maps

Author: San Ling
Journal: Proc. Amer. Math. Soc. 126 (1998), 3201-3210
MSC (1991): Primary 11G18, 14H40
MathSciNet review: 1469421
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Abstract: For an integer $M>1$ and a prime $p \geq 5$ not dividing $M$, we study the kernel of the degeneracy map $\Phi _{Mp,p}^r \rightarrow \Phi _{Mp^r, p}$, where $\Phi _{Mp,p}$ and $\Phi _{Mp^r, p}$ are the component groups of $J_0(Mp)$ and $J_0(Mp^r)$, respectively. This is then used to determine the kernel of the degeneracy map $J_0(Mp)^2 \rightarrow J_0(Mp^2)$ when $J_0(M) =0$. We also compute the group structure of $\Phi _{Mp^2, p}$ in some cases.

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  • P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1973, pp. 143–316. Lecture Notes in Math., Vol. 349 (French). MR 0337993
  • Bas Edixhoven, L’action de l’algèbre de Hecke sur les groupes de composantes des jacobiennes des courbes modulaires est “Eisenstein”, Astérisque 196-197 (1991), 7–8, 159–170 (1992) (French, with English summary). Courbes modulaires et courbes de Shimura (Orsay, 1987/1988). MR 1141457
  • Bas Edixhoven, Minimal resolution and stable reduction of $X_0(N)$, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 1, 31–67 (English, with French summary). MR 1056773
  • Nicholas M. Katz, Galois properties of torsion points on abelian varieties, Invent. Math. 62 (1981), no. 3, 481–502. MR 604840, DOI
  • Nicholas M. Katz and Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR 772569
  • San Ling, Congruences between cusp forms and the geometry of Jacobians of modular curves, Math. Ann. 295 (1993), no. 1, 111–133. MR 1198844, DOI
  • San Ling, Shimura subgroups and degeneracy maps, J. Number Theory 54 (1995), no. 1, 39–59. MR 1352635, DOI
  • S. Ling, On the Q-rational cuspidal subgroup and the component group of $J_0(p^r)$. Israel J. Math. 99 (1997), 29–54.
  • San Ling and Joseph Oesterlé, The Shimura subgroup of $J_0(N)$, Astérisque 196-197 (1991), 6, 171–203 (1992) (English, with French summary). Courbes modulaires et courbes de Shimura (Orsay, 1987/1988). MR 1141458
  • Dino J. Lorenzini, Torsion points on the modular Jacobian $J_0(N)$, Compositio Math. 96 (1995), no. 2, 149–172. MR 1326710
  • B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186 (1978). With an appendix by Mazur and M. Rapoport. MR 488287
  • Michel Raynaud, Jacobienne des courbes modulaires et opérateurs de Hecke, Astérisque 196-197 (1991), 9–25 (1992) (French, with English summary). Courbes modulaires et courbes de Shimura (Orsay, 1987/1988). MR 1141454
  • Kenneth A. Ribet, Congruence relations between modular forms, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 503–514. MR 804706
  • K. A. Ribet, On modular representations of ${\rm Gal}(\overline {\bf Q}/{\bf Q})$ arising from modular forms, Invent. Math. 100 (1990), no. 2, 431–476. MR 1047143, DOI
  • Kenneth A. Ribet, On the component groups and the Shimura subgroup of $J_0(N)$, Séminaire de Théorie des Nombres, 1987–1988 (Talence, 1987–1988) Univ. Bordeaux I, Talence, [1988?], pp. Exp. No. 6, 10. MR 993107

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Additional Information

San Ling

Keywords: Modular curves, Jacobians, component groups, degeneracy maps
Received by editor(s): April 7, 1997
Additional Notes: It is a pleasure to thank Bas Edixhoven for patiently correcting the author’s initial erroneous understanding of some concepts and for the content of §2.1.
Communicated by: David E. Rohrlich
Article copyright: © Copyright 1998 American Mathematical Society