×
Hida, Haruzo

Galois representations into \(\text{GL}_2(\mathbb Z_p[[X]])\) attached to ordinary cusp forms. (English) Zbl 0612.10021

Invent. Math. 85, 545-613 (1986).
Let \(p\) be a prime \(\geq 5\), \(N\) an integer prime to \(p\), \(\Gamma\) the topological group \(1+p{\mathbb Z}_ p\), and \(\Omega\) a \(p\)-adic completion of an algebraic closure of \({\mathbb Q}_ p\). The author uses the study of the parabolic cohomology groups of \(\Gamma_ 1(Np^ r)\) to obtain results about Galois representations attached to cusp forms. Beginning with a study of the ordinary part \(h^ 0(N,{\mathbb Z}_ p)\) of the universal Hecke algebra (generated by the Hecke operators regarded as endomorphisms of the space of \(p\)-adic cusp forms of level \(N\)) the author proves the following theorem concerning \(q\)-expansions: Let \(\Lambda ={\mathbb Z}_ p[[X]]\) be the one variable Iwasawa algebra, and fix a non-trivial \(\Lambda\)-algebra homomorphism \(\lambda: h^ 0(N,{\mathbb Z})\to \Lambda\). The image under \(\lambda\) of the formal power series \(\sum^{\infty}_{n=1}T(n)q^ n\) is, when evaluated at \(X=\varepsilon (u)u^ k-1\) \((u=1+p)\), the complex \(q\)-expansion of a common eigenform \(f_{k,\varepsilon}\) for all \(T(n)\) in \(S_ k(\Gamma_ 1(p^ r))\) where \(\varepsilon: \Gamma\to \Omega^*\) is of finite order, \(\text{ker}\;\varepsilon=1+p^ r {\mathbb Z}_ p\), and \(k\) is an integer \(\geq 2\). Moreover, \(f_{k,\varepsilon}\) is not the twist by any Dirichlet character of a form of lower level.
The author then shows that one can associate to \(\lambda\) a unique Galois representation \(\pi(\lambda): \text{Gal}(\bar{\mathbb Q}/\mathbb Q)\to \text{GL}_ 2(\Lambda)\) such that the reduction of \(\pi(\lambda)\) is equivalent to the irreducible \(\text{Gal}(\bar{\mathbb Q}/\mathbb Q)\)-representation \(\pi(f_{k,\varepsilon})\) into \(\text{GL}_ 2(\Omega)\) associated to \(f_{k,\varepsilon}\) (by Eichler-Shimura for \(f=2\), and by Deligne for \(k>2\)).
The paper ends with a study of the connections between the special values of the \(L\)-function associated to the 3-dimensional subrepresentation contained in \(\pi (f_{k,\varepsilon})\otimes {\tilde \pi}(f_{k,\varepsilon})\) (where \({\tilde \pi}\) is the contragredient representation of \(\pi\)), and the characteristic power series of a certain Iwasawa module that is associated to \(\lambda\).
Reviewer: S.Kamienny

Cited in 17 Reviews
Cited in 171 Documents

MSC:

11F80 Galois representations
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F33 Congruences for modular and \(p\)-adic modular forms
11F11 Holomorphic modular forms of integral weight
11F25 Hecke-Petersson operators, differential operators (one variable)
11R23 Iwasawa theory

Keywords:

parabolic cohomology groups; Galois representations; cusp forms; Hecke algebra; Hecke operators; \(p\)-adic cusp forms; \(q\)-expansions; Iwasawa algebra; special values of the \(L\)-function
Cite Review PDF
Full Text: DOI EuDML

References:

[1] Bourbaki, N.: Commutative algebra. Paris: Hermann 1972 · Zbl 0279.13001
[2] Casselman, W.: On some results of Atkin and Lehner. Math. Ann.201, 301-304 (1973) · Zbl 0239.10015 · doi:10.1007/BF01428197
[3] Curtis, C.W., Reiner, I.: Representation theory of finite groups and associative algebras. Pure and Applied Math.11, New York: Interscience 1962 · Zbl 0131.25601
[4] Deligne, P.: Formes modulaires et représentationsl-adiques. Sém. Bourbaki. Exp. 355 (1969)
[5] Deligne, P., Serre, J.-P.: Formes modulaires de poids 1. Ann. Sci. Ec. Norm. Super. 4ème série,7, 507-530 (1974) · Zbl 0321.10026
[6] Doi, K., Miyake, T.: Automorphic forms and number theory. (in Japanese), Tokyo: Kinokuniya Shoten 1976 · Zbl 0466.10012
[7] Gelbart, S., Jacquet, H.: A relation between automorphic representations of GL(2) and GL(3). Ann. Sci. Ec. Norm. Super, 4ème série11, 471-542 (1978) · Zbl 0406.10022
[8] Hecke, E.: Theorie der Eisensteinschen Reihen höherer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik. Abh. Math. Semin. Hamb.5, 199-224 (1927) · JFM 53.0345.02 · doi:10.1007/BF02952521
[9] 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
[10] 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
[11] Hida, H.: Kummer’s criterion for the special values of HeckeL-functions of imaginary quadratic fields and congruences among cusp forms. Invent. Math.66, 415-459 (1982) · Zbl 0485.10019 · doi:10.1007/BF01389222
[12] Hida, H.: Ap-adic measure attached to the zeta functions associated with two elliptic modular forms, I. Invent. Math.79, 159-195 (1985) · Zbl 0573.10020 · doi:10.1007/BF01388661
[13] Hida, H.: Iwasawa modules attached to congruences of cusp forms. (To appear in Ann. Sci. Ec. Norm. Super.) · Zbl 0607.10022
[14] Hida, H.: Congruences of cusp forms and Hecke algebras. Sém. Théorie des Nombres, Paris, 1983-84
[15] Langlands, R.P.: Modular forms andl-adic representations. In: Modular functions of one variable, II, Lect. Notes Math.349, pp. 361-500. Berlin-Heidelberg-New York: Springer 1973
[16] Mazur, B., Wiles, A.: Class fields of abelian extensions ofQ. Invent. Math.76, 179-330 (1984) · Zbl 0545.12005 · doi:10.1007/BF01388599
[17] Ohta, M.: Onl-adic representations attached to automorphic forms. Jap. J. Math.8 (new series), 1-47 (1982) · Zbl 0505.10012
[18] Ribet, K.A.: Galois representations attached to eigenforms with Nehentypus. In: Modular functions of one variable, V, Lect. Notes Math.601, pp. 17-52. Berlin-Heidelberg-New York: Springer 1977
[19] Serre, J.-P., Tate, J.: Good reduction of abelian varieties. Ann. Math.88, 492-517 (1968) · Zbl 0172.46101 · doi:10.2307/1970722
[20] Shimura, G.: Sur les intégrales attachées aux formes automorphes. J. Math. Soc. Japan11, 291-311 (1959) · Zbl 0090.05503 · doi:10.2969/jmsj/01140291
[21] 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
[22] Shimura, G.: On the holomorphy of certain Dirichlet series. Proc. Lond. Math. Soc.31, 79-98 (1975) · Zbl 0311.10029 · doi:10.1112/plms/s3-31.1.79
[23] Shimura, G.: The periods of certain automorphic forms of arithmetic type. J. Fac. Sci. Univ. Tokyo, Sec. IA,28, 605-632 (1982) · Zbl 0499.10027
[24] Shimura, G.: Anl-adic method in the theory of automorphic forms. (Unpublished (1968)) · Zbl 0183.25402
[25] Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Tokyo: Iwanami Shoten and Princeton Univ. Press 1971 · Zbl 0221.10029
[26] 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
[27] Tate, J.:p-divisible groups. In: Proceeding of a Conference on local fields, pp. 158-183. Berlin-Heidelberg-New York: Springer 1967 · Zbl 0157.27601
[28] Hida, H.: Hecke algebras for GL(1) and GL(2). Séminaire de Théorie des Nombres, Paris, 1984-85
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.