Duke Mathematical Journal

Congruences between cusp forms: the $(p,p)$ case

Chandrashekhar Khare

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Source
Duke Math. J. Volume 80, Number 3 (1995), 631-667.

Dates
First available: 19 February 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1077246289

Mathematical Reviews number (MathSciNet)
MR1412447

Zentralblatt MATH identifier
0857.11021

Digital Object Identifier
doi:10.1215/S0012-7094-95-08022-3

Subjects
Primary: 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50]
Secondary: 11F32: Modular correspondences, etc. 11F85: $p$-adic theory, local fields [See also 14G20, 22E50]

Citation

Khare, Chandrashekhar. Congruences between cusp forms: the ( p , p ) case. Duke Mathematical Journal 80 (1995), no. 3, 631--667. doi:10.1215/S0012-7094-95-08022-3. http://projecteuclid.org/euclid.dmj/1077246289.


Export citation

References

  • [AL] A. Atkin and J. Lehner, Hecke operators on $\Gamma \sb0(m)$, Math. Ann. 185 (1970), 134–160.
  • [ALi] A. Atkin and W. Li, Twists of newforms and pseudo-eigenvalues of $W$-operators, Invent. Math. 48 (1978), no. 3, 221–243.
  • [AS] A. Ash and G. Stevens, Modular forms in characteristic $l$ and special values of their $L$-functions, Duke Math. J. 53 (1986), no. 3, 849–868.
  • [BLR] N. Boston, H. W. Lenstra, and K. Ribet, Quotients of group rings arising from two-dimensional representations, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 4, 323–328.
  • [C] H. Carayol, Sur les représentations galoisiennes modulo $l$ attachées aux formes modulaires, Duke Math. J. 59 (1989), no. 3, 785–801.
  • [C1] H. Carayol, Sur les représentations $l$-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 409–468.
  • [Cr] J. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992.
  • [DT] F. Diamond and R. Taylor, Nonoptimal levels of mod $l$ modular representations, Invent. Math. 115 (1994), no. 3, 435–462.
  • [E] B. Edixhoven, The weight in Serre's conjectures on modular forms, Invent. Math. 109 (1992), no. 3, 563–594.
  • [G] F. Gouvêa, Arithmetic of $p$-adic modular forms, Lecture Notes in Mathematics, vol. 1304, Springer-Verlag, Berlin, 1988.
  • [Ge] S. Gelbart, Automorphic forms on adèle groups, Annals of Mathematics Studies, vol. 83, Princeton University Press, Princeton, N.J., 1975.
  • [Gr] B. Gross, A tameness criterion for Galois representations associated to modular forms (mod $p$), Duke Math. J. 61 (1990), no. 2, 445–517.
  • [H] H. Hida, Galois representations into $\rm GL\sb 2(\bf Z\sb p[[X]])$ attached to ordinary cusp forms, Invent. Math. 85 (1986), no. 3, 545–613.
  • [H1] H. Hida, Modular $p$-adic $L$ functions and $p$-adic Hecke algebras, to appear in Sugaku Expositions.
  • [H2] H. Hida, Geometric modular forms, 1992, CIMPA Summer School.
  • [J] N. Jochnowitz, A study of the local components of the Hecke algebra mod $l$, Trans. Amer. Math. Soc. 270 (1982), no. 1, 253–267.
  • [K] C. Khare, Congruences between cusp forms, Ph.D. thesis, California Institute of Technology, 1995.
  • [Ka] N. Katz, Higher congruences between modular forms, Ann. of Math. (2) 101 (1975), 332–367.
  • [La] S. Lang, Introduction to modular forms, Springer-Verlag, Berlin, 1976.
  • [Li] S. Ling, Congruences between cusp forms and the geometry of Jacobians of modular curves, Math. Ann. 295 (1993), no. 1, 111–133.
  • [LO] S. Ling and J. Oesterlé, The Shimura subgroup of $J\sb 0(N)$, Astérisque (1991), no. 196-197, 6, 171–203 (1992).
  • [M] B. Mazur, Deforming Galois representations, Galois groups over $\bf Q$ (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 385–437.
  • [M1] B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 33–186 (1978).
  • [MR] B. Mazur and K. Ribet, Two-dimensional representations in the arithmetic of modular curves, Astérisque (1991), no. 196-197, 6, 215–255 (1992).
  • [MW] B. Mazur and A. Wiles, Class fields of abelian extensions of $\bf Q$, Invent. Math. 76 (1984), no. 2, 179–330.
  • [Mi] J. Milne, Arithmetic duality theorems, Perspectives in Mathematics, vol. 1, Academic Press Inc., Boston, MA, 1986.
  • [R] K. Ribet, Congruence relations between modular forms, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984, pp. 503–514.
  • [R1] K. Ribet, On modular representations of $\rm Gal(\overline\bf Q/\bf Q)$ arising from modular forms, Invent. Math. 100 (1990), no. 2, 431–476.
  • [R2] K. Ribet, Report on mod $l$ representations of $\rm Gal(\overline\bf Q/\bf Q)$, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 639–676.
  • [R3] K. Ribet, Raising the levels of modular representations, Séminaire de Théorie des Nombres, Paris 1987–88, Progr. Math., vol. 81, Birkhäuser Boston, Boston, MA, 1990, pp. 259–271.
  • [Ra] M. Raynaud, Schémas en groupes de type $(p,\dots, p)$, Bull. Soc. Math. France 102 (1974), 241–280.
  • [S] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo, 1971.
  • [Se] J.-P. Serre, Sur les représentations modulaires de degré $2$ de $\rm Gal(\overline\bf Q/\bf Q)$, Duke Math. J. 54 (1987), no. 1, 179–230.
  • [Se1] J.-P. Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331.
  • [Se2] J.-P. Serre, Trees, Springer-Verlag, Berlin, 1980.
  • [Se3] J.-P. Serre, Le problème des groupes de congruence pour SL2, Ann. of Math. (2) 92 (1970), 489–527.
  • [W] A. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551.

See also