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

HTML articles powered by AMS MathViewer

- 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
- PDF | Request permission

## 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