## Duke Mathematical Journal

### Modular forms in characteristic $\ell$ and special values of their $L$ -functions

#### Article information

Source
Duke Math. J., Volume 53, Number 3 (1986), 849-868.

Dates
First available in Project Euclid: 20 February 2004

https://projecteuclid.org/euclid.dmj/1077305204

Digital Object Identifier
doi:10.1215/S0012-7094-86-05346-9

Mathematical Reviews number (MathSciNet)
MR860675

Zentralblatt MATH identifier
0618.10026

Subjects
Primary: 11F75: Cohomology of arithmetic groups

#### Citation

Ash, Avner; Stevens, Glenn. Modular forms in characteristic $\ell$ and special values of their $L$ -functions. Duke Math. J. 53 (1986), no. 3, 849--868. doi:10.1215/S0012-7094-86-05346-9. https://projecteuclid.org/euclid.dmj/1077305204

#### References

• [An] A. N. Andrianov, Multiplicative arithmetic of Siegel's modular forms, Uspekhi Mat. Nauk 34 (1979), no. 1(205), 67–135.
• [A-S] A. Ash and G. Stevens, Cohomology of arithmetic groups and congruences between systems of Hecke eigenvalues, J. Reine Angew. Math. 365 (1986), 192–220.
• [D] P. Deligne, Formes modulaires et representations de $\ell$-adiques, Lecture Notes in Math., vol. 179, 1971, Sem. Bourbaki, pp. 139–186.
• [Di] Leonard Eugene Dickson, A fundamental system of invariants of the general modular linear group with a solution of the form problem, Trans. Amer. Math. Soc. 12 (1911), no. 1, 75–98.
• [D-O] K. Doi and M. Ohta, On some congruences between cusp forms on $\Gamma \sb0(N)$, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Springer, Berlin, 1977, 91–105. Lecture Notes in Math., Vol. 601.
• [E] M. Eichler, Eine Verallgemeinerung der Abelschen Integrale, Math. Z. 67 (1957), 267–298.
• [Ha] K. Haberland, Perioden von Modulformen einer Variabler and Gruppencohomologie. I, II, III, Math. Nachr. 112 (1983), 245–282, 283–295, 297–315.
• [H1] H. Hida, On congruence divisors of cusp forms as factors of the special values of their zeta functions, Invent. Math. 64 (1981), no. 2, 221–262.
• [H2] H. Hida, Congruence of cusp forms and special values of their zeta functions, Invent. Math. 63 (1981), no. 2, 225–261.
• [H3] H. Hida, Kummer's criterion for the special values of Hecke $L$-functions of imaginary quadratic fields and congruences among cusp forms, Invent. Math. 66 (1982), no. 3, 415–459.
• [J] N. Jochnowitz, Congruences between systems of eigenvalues of modular forms, Trans. Amer. Math. Soc. 270 (1982), no. 1, 269–285.
• [K-P-S] M. Kuga, W. Parry, and C. H. Sah, Group cohomology and Hecke operators, Manifolds and Lie groups (Notre Dame, Ind., 1980), Progr. Math., vol. 14, Birkhäuser Boston, Mass., 1981, pp. 223–266.
• [M1] J. Manin, Periods of parabolic forms and $p$-adic Hecke series, Math. USSR Sbornik 21 (1973), 371–393.
• [M2] J. Manin, The values of $p$-adic Hecke series at integer points of the critical strip, Math. USSR Sbornik 22 (1974), 631–637.
• [Mz] B. Mazur, On the arithmetic of special values of $L$ functions, Invent. Math. 55 (1979), no. 3, 207–240.
• [O] M. Ohta, On $l$-adic representations attached to automorphic forms, Japan. J. Math. (N.S.) 8 (1982), no. 1, 1–47.
• [Ra] M. Razar, Dirichlet series and Eichler cohomology, preprint.
• [R1] K. Ribet, Mod $p$ Hecke operators and congruences between modular forms, Invent. Math. 71 (1983), no. 1, 193–205.
• [R2] K. Ribet, On $l$-adic representations attached to modular forms, Invent. Math. 28 (1975), 245–275.
• [Se1] J. P. Serre, Formes modulaires et fonctions zêta $p$-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), Springer, Berlin, 1973, 191–268. Lecture Notes in Math., Vol. 350.
• [Se2] J. P. Serre, Letter to J.-M. Fontaine, 1979.
• [S1] G. Shimura, An $\ell$-adic method in the theory of automorphic forms, Unpublished, 1968.
• [S2] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo, 1971.
• [St1] G. Stevens, The cuspidal group and special values of $L$-functions, Trans. Amer. Math. Soc. 291 (1985), no. 2, 519–550.
• [St2] G. Stevens, Arithmetic on modular curves, Progress in Mathematics, vol. 20, Birkhäuser Boston Inc., Boston, MA, 1982.
• [Sw-D]1 H. P. F. Swinnerton-Dyer, On $l$-adic representations and congruences for coefficients of modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), Springer, Berlin, 1973, 1–55. Lecture Notes in Math., Vol. 350.
• [Sw-D]2 H. P. F. Swinnerton-Dyer, On $l$-adic representations and congruences for coefficients of modular forms. II, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Springer, Berlin, 1977, 63–90. Lecture Notes in Math., Vol. 601.