On Ramanujan congruences for modular forms of integral and half-integral weights
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- by B. Datskovsky and P. Guerzhoy
- Proc. Amer. Math. Soc. 124 (1996), 2283-2291
- DOI: https://doi.org/10.1090/S0002-9939-96-03334-5
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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.References
- Henri Cohen, Sums involving the values at negative integers of $L$-functions of quadratic characters, Math. Ann. 217 (1975), no. 3, 271–285. MR 382192, DOI 10.1007/BF01436180
- Pierre Deligne and Jean-Pierre Serre, Formes modulaires de poids $1$, Ann. Sci. École Norm. Sup. (4) 7 (1974), 507–530 (1975) (French). MR 379379, DOI 10.24033/asens.1277
- Guerzhoy, P., On Ramanujan congruences between special values of Hecke and Dirichlet $L$-functions, preprint.
- Nicholas M. Katz, Higher congruences between modular forms, Ann. of Math. (2) 101 (1975), 332–367. MR 417059, DOI 10.2307/1970994
- Neal Koblitz, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1984. MR 766911, DOI 10.1007/978-1-4684-0255-1
- Neal Koblitz, $p$-adic congruences and modular forms of half integer weight, Math. Ann. 274 (1986), no. 2, 199–220. MR 838465, DOI 10.1007/BF01457070
- Winfried Kohnen, Modular forms of half-integral weight on $\Gamma _{0}(4)$, Math. Ann. 248 (1980), no. 3, 249–266. MR 575942, DOI 10.1007/BF01420529
- Yoshitaka Maeda, A congruence between modular forms of half-integral weight, Hokkaido Math. J. 12 (1983), no. 1, 64–73. MR 689257, DOI 10.14492/hokmj/1381757792
- Ju. I. Manin, Periods of cusp forms, and $p$-adic Hecke series, Mat. Sb. (N.S.) 92(134) (1973), 378–401, 503 (Russian). MR 0345909
- Ramanujan, S., On certain arithmetic functions, Trans. Cambridge Phil. Soc. 22 (1916), 159–184.
- Kenneth A. Ribet, On $l$-adic representations attached to modular forms, Invent. Math. 28 (1975), 245–275. MR 419358, DOI 10.1007/BF01425561
- Kenneth A. Ribet, A modular construction of unramified $p$-extensions of $Q(\mu _{p})$, Invent. Math. 34 (1976), no. 3, 151–162. MR 419403, DOI 10.1007/BF01403065
- 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
- Jean-Pierre Serre, Congruences et formes modulaires [d’après H. P. F. Swinnerton-Dyer], Séminaire Bourbaki, 24ème année (1971/1972), Lecture Notes in Math., Vol. 317, Springer, Berlin, 1973, pp. Exp. No. 416, pp. 319–338 (French). MR 0466020
- Jean-Pierre Serre, Divisibilité des coefficients des formes modulaires de poids entier, C. R. Acad. Sci. Paris Sér. A 279 (1974), 679–682 (French). MR 382172
- Goro Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440–481. MR 332663, DOI 10.2307/1970831
- Max Zorn, Continuous groups and Schwarz’ lemma, Trans. Amer. Math. Soc. 46 (1939), 1–22. MR 53, DOI 10.1090/S0002-9947-1939-0000053-7
- S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495–503. MR 0000089, DOI 10.1007/BF03046994
- Samuel S. Wagstaff Jr., The irregular primes to $125000$, Math. Comp. 32 (1978), no. 142, 583–591. MR 491465, DOI 10.1090/S0025-5718-1978-0491465-4
Bibliographic Information
- B. Datskovsky
- Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
- Email: bdats@euclid.math.temple.edu
- 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
- Email: pasha@techunix.technion.ac.il, pasha@euklid.math.uni-mannheim.de
- 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
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 2283-2291
- MSC (1991): Primary 11F33; Secondary 11F30, 11F32, 11F37
- DOI: https://doi.org/10.1090/S0002-9939-96-03334-5
- MathSciNet review: 1327004