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Congruences of cusp forms and special values of their zeta functions. (English) Zbl 0459.10018


MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F33 Congruences for modular and \(p\)-adic modular forms
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14F20 Étale and other Grothendieck topologies and (co)homologies

References:

[1] Asai, T.: On the Fourier coefficients of automorphic forms at various cusps and some application to Rankin’s convolution, J. Math. Soc. Japan28, 48-61 (1976) · Zbl 0313.10026 · doi:10.2969/jmsj/02810048
[2] Atkin, A.O.L., Lehner, J.: Hecke operators on ?0(m), Math. Ann.185, 134-160 (1970) · doi:10.1007/BF01359701
[3] Bourbaki, N.: Algebra 9 (sesquilinear forms and quadratic forms), Paris: Hermann 1959 · Zbl 0102.25503
[4] Bourbaki, N.: Commutative algebra, Paris: Hermann 1972 · Zbl 0279.13001
[5] Bredon, G.E.: Sheaf theory. New York: McGraw-Hill 1967 · Zbl 0158.20505
[6] Damerell, R.M.:L-functions of elliptic curves with complex multiplication I, II, Acta Arith.17, 287-301 (1970);19, 311-317 (1971) · Zbl 0209.24603
[7] Deligne, P.: Formes modulaires et representationsl-adiques, Sém. Bourbaki, exp. 355, fév. 1969
[8] Deligne, P., Serre, J-P.: Formes modulaires de poids 1, Ann. Sci. École Norm. Sup. 4e série, tome7, 507-530 (1974) · Zbl 0321.10026
[9] Doi, K., Hida, H.: On a certain congruence of cusp forms and the special values of their Dirichlet series, unpublished (1979)
[10] Doi, K., Miyake, T.: Automorphic forms and number theory (in Japanese). Tokyo: Kinokuniya Shoten 1976 · Zbl 0466.10012
[11] Doi, K., Ohta, M.: On some congruences between cusp forms on ?0(N). In: Modular functions of one variable, V, Lectures Notes in Mathematics, 601, pp. 91-105. Berlin-Heidelberg-New York: Springer 1977 · Zbl 0361.10023
[12] Eckmann, B.: Cohomology of groups and transfer, Ann. of Math.58, 481-493 (1953) · Zbl 0052.02002 · doi:10.2307/1969749
[13] Hida, H.: On abelian varieties with complex multiplication as factors of the abelian variety attached to Hilbert modular forms. Jap. J. Math.5, 157-208 (1979) · Zbl 0422.14026
[14] Hurwitz, A.: Über die Entwicklungskoeffizienten der lemniskatishen Funktionen. Math. Ann.51, 196-226 (1899), (Werke II, 342-373) · JFM 29.0385.02 · doi:10.1007/BF01453637
[15] Labesse, J.-P., Langlands, R.P.:L-indistinguishability forSL(2). Can. J. Math.31, 726-785 (1979) · Zbl 0421.12014 · doi:10.4153/CJM-1979-070-3
[16] Langlands, R.P.: Modular forms andl-adic representations. In: Modular functions of one variable, II, Lecture Notes in Mathematics, 349, pp. 361-500. Berlin-Heidelberg-New York: Springer 1973
[17] Matter, K.: Die den Bernoullischen Zahlen analogen Zahlen im Körper der dritten Einheitenwurzeln, Vierteljahrsschrift d. Naturf. Ges. Zürich45, 238-269 (1900)
[18] Mazur, B.: Rational isogenies of prime degree. Inventiones Math.44, 129-162 (1978) · Zbl 0386.14009 · doi:10.1007/BF01390348
[19] Mazur, B.: On the arithmetic of special values ofL-functions. Inventiones Math.55, 207-240 (1979) · Zbl 0426.14009 · doi:10.1007/BF01406841
[20] Mazur, B., Swinnerton-Dyer, P.: Arithmetic of Weil curves. Inventiones Math.25, 1-61 (1974) · Zbl 0281.14016 · doi:10.1007/BF01389997
[21] Milne, J.S.: Étale Cohomology, Princeton University Press 1980 · Zbl 0433.14012
[22] Miyake, T.: On automorphic forms onGL 2 and Hecke operators. Ann. of Math.94, 174-189 (1971) · Zbl 0215.37301 · doi:10.2307/1970741
[23] Ohta, M.: Onl-adic representations attached to automorphic forms, preprint
[24] Serre, J-P.: Cohomologie des groupes discrets. In: Prospects in Mathematics. Annals of Mathematics Studies, 70, pp. 77-170, Princeton University Press 1971 · Zbl 0229.57016
[25] Shimura, G.: Sur les intégrales attachées aux formes automorphes. J. Math. Soc. Japan11, 291-311 (1959) · Zbl 0090.05503 · doi:10.2969/jmsj/01140291
[26] Shimura, G.: Introduction to the arithmetic theory of automorphic functions, Iwanami Shoten and Princeton University Press 1971 · Zbl 0221.10029
[27] Shimura, G.: On elliptic curves with complex multiplication as factors of the jacobians of modular function fields. Nagoya Math. J.43, 199-208 (1971) · Zbl 0225.14015
[28] Shimura, G.: On the factors of the jacobian variety of a modular function field. J. Math. Soc. Japan25, 523-544 (1973) · Zbl 0266.14017 · doi:10.2969/jmsj/02530523
[29] Shimura, G.: On the holomorphy of certain Dirichlet series. Proc. London Math. Soc.31, 79-98 (1975) · Zbl 0311.10029 · doi:10.1112/plms/s3-31.1.79
[30] Shimura, G.: The special values of the zeta functions associated with cusp forms. Comm. pure appl. Math.29, 783-804 (1976) · Zbl 0348.10015 · doi:10.1002/cpa.3160290618
[31] Shimura, G.: On the periods of modular forms. Math. Ann.229, 211-221 (1977) · Zbl 0363.10019 · doi:10.1007/BF01391466
[32] Shimura, G.: The special values of the zeta functions associated with Hilbert modular forms. Duke Math. J.45, 637-679 (1978) · Zbl 0394.10015 · doi:10.1215/S0012-7094-78-04529-5
[33] Sturm, J.: Special values of zeta functions and Eisenstein series of half integral weight. Amer. J.102, 219-240 (1980) · Zbl 0433.10015
[34] [SGA 4III], Théorie des Topos et Cohomologie Etale des Schémas, Tome 3. Lecture Notes in Mathematics, 305. Berlin-Heidelberg-New York: Springer 1973
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