Congruences for the Ramanujan function and generalized class numbers
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Abstract:
The Ramanujan $\tau$-function satisfies well-known congruences modulo the so-called exceptional prime numbers $2,3,5,7,23,691$. In this paper we prove new congruences related to the irregular primes $131$ and $593$, involving generalized class numbers. As an application we obtain distribution results. We obtain a new proof of the famous $691$ congruence and congruences of the related Rankin L-funtion.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
- P. Deligne: Formes modulaires et representations l-adiques. Seminaire Bourbaki No. 355 (1969).
- Martin Eichler and Don Zagier, The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781735, DOI 10.1007/978-1-4684-9162-3
- Paul B. Garrett, Decomposition of Eisenstein series: Rankin triple products, Ann. of Math. (2) 125 (1987), no. 2, 209–235. MR 881269, DOI 10.2307/1971310
- Paul B. Garrett, On the arithmetic of Siegel-Hilbert cuspforms: Petersson inner products and Fourier coefficients, Invent. Math. 107 (1992), no. 3, 453–481. MR 1150599, DOI 10.1007/BF01231899
- G. H. Hardy, Ramanujan. Twelve lectures on subjects suggested by his life and work, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1940. MR 0004860
- B. Heim: On the Spezialschar of Maass. Preprint, submitted 2006
- Hans Maass, Die Fourierkoeffizienten der Eisensteinreihen zweiten Grades, Mat.-Fys. Medd. Danske Vid. Selsk. 34 (1964), no. 7, 25 pp. (1964) (German). MR 171758
- Ken Ono, Congruences on the Fourier coefficients of modular forms on $\Gamma _0(N)$, The Rademacher legacy to mathematics (University Park, PA, 1992) Contemp. Math., vol. 166, Amer. Math. Soc., Providence, RI, 1994, pp. 93–105. MR 1284053, DOI 10.1090/conm/166/01643
- Jean-Pierre Serre, Abelian $l$-adic representations and elliptic curves, W. A. Benjamin, Inc., New York-Amsterdam, 1968. McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute. MR 0263823
- S. Ramanujan: On certain arithmetical functions. Trans. Cambridge Phil. Soc. 22 (1916), 159-184.
- J.-P. Serre: Congruences et formes modulaires (d’apres Swinnerton-Dyer). Seminaire Bourbaki No. 416 (1971).
- D. Zagier, Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields, Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Lecture Notes in Math., Vol. 627, Springer, Berlin, 1977, pp. 105–169. MR 0485703
Additional Information
- Bernhard Heim
- Affiliation: Max-Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
- Email: heim@mpim-bonn.mpg.de
- Received by editor(s): November 13, 2007
- Received by editor(s) in revised form: January 9, 2008
- Published electronically: May 20, 2008
- © Copyright 2008 American Mathematical Society
- Journal: Math. Comp. 78 (2009), 431-439
- MSC (2000): Primary 11F33, 11F67, 11F80; Secondary 11Y70
- DOI: https://doi.org/10.1090/S0025-5718-08-02136-4
- MathSciNet review: 2448715