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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On Ramanujan congruences
for modular forms of integral
and half-integral weights

Authors: B. Datskovsky and P. Guerzhoy
Journal: Proc. Amer. Math. Soc. 124 (1996), 2283-2291
MSC (1991): Primary 11F33; Secondary 11F30, 11F32, 11F37
MathSciNet review: 1327004
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Abstract: In 1916 Ramanujan observed a remarkable congruence: $\tau (n)\equiv \sigma _{11}(n) \quad \bmod \, 691$. The modern point of view is to interpret the Ramanujan congruence as a congruence between the Fourier coefficients of the unique normalized cusp form of weight $12$ and the Eisenstein series of the same weight modulo the numerator of the Bernoulli number $B_{12}$. In this paper we give a simple proof of the Ramanujan congruence and its generalizations to forms of higher integral and half-integral weights.

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  • [1] Cohen, H., Sums involving the values at negative integers of $L$-functions of quadratic characters, Math. Ann. 217 (1975), 271--285. MR 52:3080
  • [2] Deligne, P. and Serre, J.-P., Formes modulaires de poids 1, Ann. Sci. Ecole Norm. Sup. 7 (1974), 507--530. MR 52:284
  • [3] Guerzhoy, P., On Ramanujan congruences between special values of Hecke and Dirichlet $L$-functions, preprint.
  • [4] Katz, N. M., Higher congruences between modular forms, Ann. Math. 101 (1975), 332--367. MR 54:5120
  • [5] Koblitz, N., Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, New York-Heidelberg-Berlin, 1984. MR 86c:11040
  • [6] Koblitz, N., $p$-Adic congruences and modular forms of half integer weight, Math. Ann. 274 (1986), 199--220. MR 88a:11043
  • [7] Kohnen, W., Modular forms of half-integral weight on $\Gamma _{0}(4)$, Math. Ann. 248 (1980), 249--266. MR 81j:10030
  • [8] Maeda, Y., A congruence between modular forms of half-integral weight, Hokkaido Math. J. 12 (1983), 64--73. MR 84e:10036
  • [9] Manin, Y. I., Periods of parabolic forms and $p$-adic Hecke series, Math. USSR Sbornik 92 (1973), 378--401; English transl., Math. USSR Sbornik 21 (1973), 371--393. MR 49:10638
  • [10] Ramanujan, S., On certain arithmetic functions, Trans. Cambridge Phil. Soc. 22 (1916), 159--184.
  • [11] Ribet, K. A., On $l$-adic representations attached to modular forms, Invent. Math. 28 (1975), 245--275. MR 54:7379
  • [12] Ribet, K.A., A modular construction of unramified $p$-extensions of $Q(\mu _{p})$, Invent. Math. 34 (1976), 151--162. MR 54:7424
  • [13] Ribet, K. A., Congruence relations between modular forms, Proceedings of the International Congress of Mathematicians (Warsaw, 1983), PWN, Warsaw, 1984, pp. 503--514. MR 87c:11045
  • [14] Serre, J.-P., Congruences et formes modulaires (d'après H. P. F. Swinnerton-Dyer), Lect. Notes Math. 317, Springer-Verlag, Berlin-Heidelberg-New York, 1973, pp. (319--338). MR 57:5904a
  • [15] Serre, J.-P., Divisibilité des coefficients des formes modulaires de poids entiers, C. R. Acad. Sci. Paris 279 (1974), 679--682. MR 52:3060
  • [16] Shimura, G., On modular forms of half integral weight, Ann. Math. 97 (1973), 440--481. MR 48:10989
  • [17] Swinnerton - Dyer, H. P. F., On $l$-adic representations and congruences for coefficients of forms, Lect. Notes Math. 350, Springer-Verlag, Berlin-Heidelberg-New York, 1973, pp. 1-- 55. MR 53:10717
  • [18] Swinnerton - Dyer, H. P. F., Congruence properties of $\tau (n)$, in Ramanujan Revisited, Proceedings of the Centenary Conference, Academic Press, San Diego-London, 1988. MR 89e:10028
  • [19] Wagstaff, S., The irregular primes to 125,000, Math. Comp. 32 (1978), 583--591. MR 58:10711

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Additional Information

B. Datskovsky
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122

P. Guerzhoy
Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, 32000 Haifa, Israel
Address at time of publication: Fakultät für Mathematik und Informatik, Universität Mannheim, D-6800 Mannheim 1, Germany

Received by editor(s): May 15, 1994
Additional Notes: The first author’s research was supported by a Fulbright fellowship.
Communicated by: Dennis A. Hejhal
Article copyright: © Copyright 1996 American Mathematical Society