zbMATH — the first resource for mathematics

Kummer’s criterion for the special values of Hecke L-functions of imaginary quadratic fields and congruences among cusp forms. (English) Zbl 0485.10019

11F11 Holomorphic modular forms of integral weight
11F33 Congruences for modular and \(p\)-adic modular forms
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
Full Text: DOI EuDML
[1] Bourbaki, N.: Commutative Algebra. Paris: Hermann 1972
[2] Brumer, A.: On the units of algebraic number fields. Matematika,14, 121-124 (1967) · Zbl 0171.01105 · doi:10.1112/S0025579300003703
[3] Coates, J., Wiles, A.: Kummer’s criterion for Hurwitz numbers. In: Proc. of Int. Symp. on Alg. Number Theory, pp. 9-22. Kyoto 1976 · Zbl 0369.12009
[4] Curtis, C.W., Reiner, I.: Representation Theory of Finite Groups and Associative Algebras. Interscience Publishers 1962 · Zbl 0131.25601
[5] Damerell, R.M.:L-functions of elliptic curves with complex multiplication I. Acta Arith., 17, 287-301 (1970); II, Acta Arith.,19, 311-317 (1971) · Zbl 0209.24603
[6] Deligne, P.: Formes modulaires et représentationsl-adiques. Sém. Bourbaki, exp. 355, fév. 1969
[7] Deligne, P., Rapoport, M.: Les schémas de modules de courbes elliptiques. In: Modular Functions of One Variable II, Lecture Notes in Mathematics, 349, pp. 143-174. Berlin, Heidelberg, New York: Springer-Verlag 1973
[8] Deligne, P., Serre, J-P.: Formes modulaires de poids 1. Ann. Sci. Ecole Norm. Sup. 4e série, t.7, 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 Unpublished
[10] Doi, K., Ohta, M.: On some congruences between cusp forms on? 0(N). In: Modular Functions of One Variable V, Lecture Notes in Mathematics, 601, pp. 91-105. Berlin, Heidelberg, New York: Springer-Verlag 1977
[11] Hida, H.: On abelian varieties with complex multiplication as factors of the jacobians of Shimura curves. Amer. J. Math.103, 727-776 (1981) · Zbl 0477.14024 · doi:10.2307/2374049
[12] Hida, H.: Congruences of cusp forms and special values of their zeta functions. Invent. Math.63, 225-261 (1981) · Zbl 0459.10018 · doi:10.1007/BF01393877
[13] Hida, H.: On congruence divisors of cusp forms as factors of the special values of their zeta functions. Invent. Math.64, 221-262 (1981) · Zbl 0472.10028 · doi:10.1007/BF01389169
[14] Iwasawa, K.: Lectures onp-adicL-functions. Annals of Mathematics Studies, 74, Princeton University Press 1972
[15] Koike, M.: A note on modular forms modp. Proc. Japan Acad. Ser. A,55, 313-315 (1979) · Zbl 0439.10017 · doi:10.3792/pjaa.55.313
[16] Mazur, B.: Rational isogenies of prime degree. Invent. Math.44, 129-162 (1978) · Zbl 0386.14009 · doi:10.1007/BF01390348
[17] Miyake, T.: On automorphic forms onGL 2 and Hecke operators. Ann. of Math.94, 174-189 (1971) · Zbl 0215.37301 · doi:10.2307/1970741
[18] Néron, A.: Modèles minimaux des variétés abéliennes sur les corps locaux et globaux. Publ. Math. I.H.E.S.21, 5-128 (1964)
[19] Ohta, M.: The representation of Galois group attached to certain finite group schemes, and its application to Shimura’s theory. In: Proc. of Int. Symp. on Alg. Number Theory, Kyoto, pp. 149-156, 1976
[20] Raynaud, M.: Schémas en groupes de type (p,...,p). Bull. Soc. Math. France102, 241-280 (1974) · Zbl 0325.14020
[21] Ribet, K.A.: A modular construction of unramifiedp-extensions ofQ(? p ). Invent. Math.34, 151-162 (1976) · Zbl 0338.12003 · doi:10.1007/BF01403065
[22] Ribet, K.A.: Galois representations attached to eigenforms with Nebentypes. In: Modular Functions of One Variable V. Lecture Notes in Mathematics, 601, pp. 18-52. Berlin, Heidelberg, New York: Springer-Verlag 1977
[23] Robert, G.: Nombres de Hurwitz et unités elliptiques. Ann. Sci. École Norm. Sup., 4e-série, t.11, 297-389 (1978)
[24] Serre, J-P.: Abelianl-Adic Representations and Elliptic Curves. W.A. Benjamin, Inc. 1968
[25] Serre, J-P., Tate, J.: Good reduction of abelian varieties. Ann. of Math.88, 492-517 (1968) · Zbl 0172.46101 · doi:10.2307/1970722
[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.: Class field over real quadratic fields and Hecke operators. Ann. of Math.95, 130-190 (1972) · Zbl 0255.10032 · doi:10.2307/1970859
[29] 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
[30] 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
[31] Shimura, G., Taniyama, Y.: Complex Multiplication of Abelian Varieties and Its Application to Number Theory. Publ. Math. Soc. Japan, No. 6, 1961 · Zbl 0112.03502
[32] Sturm, J.: Special values of zeta functions, and Eisenstein series of half integral weight. Amer. J. Math.102, 219-240 (1980) · Zbl 0433.10015 · doi:10.2307/2374237
[33] Tate, J.:p-Divisible groups. In: Proceedings of a Conference on Local Fields. Driebergen, 1966, pp. 158-183. Berlin, Heidelberg, New York: Springer-Verlag 1967
[34] Weil, A.: On a certain type of characters of the idèle class group of an algebraic number field. In: Proc. Symp. on Algebraic Number Theory, Tokyo-Nikko, pp. 1-7, 1955
[35] Wiles, A.: Modular curves and the class group ofQ(? p ). Invent. Math.58, 1-35 (1980) · Zbl 0436.12004 · doi:10.1007/BF01402272
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.