Duke Mathematical Journal

Integral Hodge theory and congruences between modular forms

Bruce W. Jordan and Ron Livné

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 2 (1995), 419-484.

Dates
First available in Project Euclid: 19 February 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1077246089

Digital Object Identifier
doi:10.1215/S0012-7094-95-08017-X

Mathematical Reviews number (MathSciNet)
MR1369399

Zentralblatt MATH identifier
0851.11032

Subjects
Primary: 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50]
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields

Citation

Jordan, Bruce W.; Livné, Ron. Integral Hodge theory and congruences between modular forms. Duke Math. J. 80 (1995), no. 2, 419--484. doi:10.1215/S0012-7094-95-08017-X. https://projecteuclid.org/euclid.dmj/1077246089


Export citation

References

  • [1] L. Barthel and R. Livné, Modular representations of $GL_2$ of a local field: the ordinary, unramified case, to appear in J. Number Theory.
  • [2] I. N. Bernstein and A. V. Zelevinskii, Representations of the group $GL(n,F),$ where $F$ is a local non-Archimedean field, Uspehi Mat. Nauk 31 (1976), no. 3(189), 5–70, Translation in Russian Math Surveys 31(3)(1976), 1–68.
  • [3] H. Brandt, Zur Zahlentheorie der Quaternionen, Jber. Deutsch. Math. Verein. 53 (1943), 23–57.
  • [4] G. E. Bredon, Equivariant cohomology theories, Lecture Notes in Mathematics, No. 34, Springer-Verlag, Berlin, 1967.
  • [5] 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.
  • [6] I. V. Cerednik, Uniformization of algebraic curves by discrete arithmetic subgroups of ${\rm PGL}\sb{2}(k\sb{w})$ with compact quotient spaces, Mat. Sb. (N.S.) 100(142) (1976), no. 1, 59–88, 165, Translation in Math. USSR Sb. 29(1976), 55–78.
  • [7] P. Deligne and J.-P. Serre, Formes modulaires de poids $1$, Ann. Sci. École Norm. Sup. (4) 7 (1974), 507–530 (1975).
  • [8] F. Diamond, Congruence primes for cusp forms of weight $k\ge 2$, Astérisque (1991), no. 196-197, 6, 205–213 (1992).
  • [9] F. Diamond and R. Taylor, Nonoptimal levels of mod $l$ modular representations, Invent. Math. 115 (1994), no. 3, 435–462.
  • [10] V. G. Drinfeld, Coverings of $p$-adic symmetric domains, Funkcional. Anal. i Priložen. 10 (1976), no. 2, 29–40, Translation in Functional Anal. Appl. 10(1976), 107–115.
  • [11] B. Eckmann, Harmonische Funktionen und Randwertaufgaben in einem Komplex, Comment. Math. Helv. 17 (1945), 240–255.
  • [12]1 M. Eichler, The basis problem for modular forms and the traces of the Hecke operators, Modular functions of one variable, I (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, 75–151. Lecture Notes in Math., Vol. 320.
  • [12]2 M. Eichler, Correction to: “The basis problem for modular forms and the traces of the Hecke operators” (Modular functions of one variable, I (Proc. Internat. Summer School, Univ. Antwerp, 1972), pp. 75–151, Lecture Notes in Math., Vol. 320, Springer, Berlin, 1973), Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1975, 145–147. Lecture Notes in Math., Vol. 476.
  • [13]1 M. Eichler, Zur Zahlentheorie der Quaternionen-Algebren, J. Reine Angew. Math. 195 (1955), 127–151 (1956).
  • [13]2 M. Eichler, Berichtigung zu der Arbeit “Zur Zahlentheorie der Quaternionen-Algebren”, J. Reine Angew. Math. 197 (1957), 220.
  • [14] H. Garland, $p$-adic curvature and the cohomology of discrete subgroups of $p$-adic groups, Ann. of Math. (2) 97 (1973), 375–423.
  • [15] S. Gelbart, Automorphic forms on adèle groups, Princeton University Press, Princeton, N.J., 1975.
  • [16] S. Gelbart and H. Jacquet, Forms of ${\rm GL}(2)$ from the analytic point of view, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 213–251.
  • [17] A. Grothendieck, Groupes de monodromie en géométrie algébrique. I, Lecture Notes in Mathematics, vol. 288, Springer-Verlag, Berlin, 1972.
  • [18] H. Hida, On $p$-adic Hecke algebras for ${\rm GL}\sb 2$ over totally real fields, Ann. of Math. (2) 128 (1988), no. 2, 295–384.
  • [19] H. Hijikata, A. K. Pizer, and T. R. Shemanske, The basis problem for modular forms on $\Gamma\sb 0(N)$, Mem. Amer. Math. Soc. 82 (1989), no. 418, vi+159.
  • [20] H. Jacquet and R. P. Langlands, Automorphic Forms on ${\rm GL}(2)$, Lecture Notes in Mathematics, vol. 114, Springer-Verlag, Berlin, 1970.
  • [21] B. W. Jordan and R. Livné, Conjecture “epsilon” for weight $k>2$, Bull. Amer. Math. Soc. (N.S.) 21 (1989), no. 1, 51–56.
  • [22] B. W. Jordan and R. Livné, Local Diophantine properties of Shimura curves, Math. Ann. 270 (1985), no. 2, 235–248.
  • [23] B. W. Jordan and R. Livné, On the Néron model of Jacobians of Shimura curves, Compositio Math. 60 (1986), no. 2, 227–236.
  • [24] S. Lang, Introduction to modular forms, Springer-Verlag, Berlin, 1976.
  • [25] B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 33–186 (1978).
  • [26] M. Raynaud, Spécialisation du foncteur de Picard, Inst. Hautes Études Sci. Publ. Math. (1970), no. 38, 27–76.
  • [27] K. A. Ribet, Bimodules and abelian surfaces, Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 359–407.
  • [28] K. A. Ribet, On the component groups and the Shimura subgroup of $J\sb 0(N)$, Séminaire de Théorie des Nombres, 1987–1988 (Talence, 1987–1988), Univ. Bordeaux I, Talence, 19??, Exp. No. 6, 10.
  • [29] K. 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.
  • [30] 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.
  • [31] J.-P. Serre, Arbres, Amalgames, $SL_{2}$, Société Mathematique de France, Paris, 1982.
  • [32] J.-P. Serre, Corps locaux, Hermann, Paris, 1968.
  • [33] J.-P. Serre, May 27 1979, Lettre to J.-M. Fontaine.
  • [34] 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.
  • [35] H. Shimizu, On zeta functions of quaternion algebras, Ann. of Math. (2) 81 (1965), 166–193.
  • [36] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo, 1971.
  • [37] R. Taylor, On Galois representations associated to Hilbert modular forms, Invent. Math. 98 (1989), no. 2, 265–280.
  • [38] J. Teitelbaum, Modular representations of ${\rm PGL}\sb 2$ and automorphic forms for Shimura curves, Invent. Math. 113 (1993), no. 3, 561–580.