Wiles, A. On ordinary \(\lambda\)-adic representations associated to modular forms. (English) Zbl 0664.10013 Invent. Math. 94, No. 3, 529-573 (1988). Let F be a totally real field, \({\mathfrak O}_ F\) its ring of integers, \({\mathfrak c}\) an ideal of \({\mathfrak O}_ F\), and \(\psi\) a character of (\({\mathfrak O}/{\mathfrak c})^*\). Let f be a primitive Hilbert modular form of weight k, level \({\mathfrak c}\), and character \(\psi\). Suppose that \(T({\mathfrak p})f=c({\mathfrak p},f)f\) for each Hecke operator \(T({\mathfrak p})\) associated to an ideal \({\mathfrak p}\) of \({\mathfrak O}\). The modular form f is said to be ordinary at the prime \(\lambda\) of \({\mathfrak O}_ f\) if for each prime \({\mathfrak p}\) dividing the norm of \(\lambda\), the equation \(x^ 2-c({\mathfrak p},f)x+\psi ({\mathfrak p})N{\mathfrak p}^{k-1}=0\) has at least one root which is a unit mod \(\lambda\). (Here \({\mathfrak O}_ f\) is the integer ring of the field generated by the c(\({\mathfrak p},f).)\) Using Hida’s theory of \(\Lambda\)-adic newforms the author shows that if f is a primitive modular form which is ordinary at \(\lambda\) then there exists a continuous \(\lambda\)-adic representation \(\rho_{\lambda}: Gal(\bar F/F)\to GL_ 2({\mathfrak O}_{\lambda})\) unramified outside \({\mathfrak c}\cdot (N\lambda)\) and such that for all primes \({\mathfrak q}\nmid {\mathfrak c}\cdot (N\lambda),\quad trace \rho_{\lambda}(Frob_{{\mathfrak q}})=c({\mathfrak q},f)\) and \(\det \rho_{\lambda}(Frob_{{\mathfrak q})=\psi ({\mathfrak q}})N{\mathfrak q}^{k-1}\) (where \(Frob_{{\mathfrak q}}\) is a \({\mathfrak q}\)-Frobenius automorphism in Gal\((\bar F/F)\)). Moreover, in weight one the image of \(\rho_{\lambda}\) is finite and lifts to a complex 2-dimensional representation. In addition, the restriction of \(\rho_{\lambda}\) to a \({\mathfrak p}\)-decomposition group is described explicitly. The existence of such representations has long been conjectured (whether or not f is ordinary). The conjecture was already known in various cases including when \([F:{\mathbb{Q}}]\) is odd. Reviewer: S.Kamienny Cited in 5 ReviewsCited in 141 Documents MSC: 11F33 Congruences for modular and \(p\)-adic modular forms 11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces Keywords:ordinary modular form; Hilbert modular form; continuous \(\lambda\)-adic representation; Frobenius automorphism PDF BibTeX XML Cite \textit{A. Wiles}, Invent. Math. 94, No. 3, 529--573 (1988; Zbl 0664.10013) Full Text: DOI EuDML OpenURL References: [1] [A] Arthur, J.: The Selberg trace formula for groups ofF-rank one. Ann. Math.100, 326-385 (1974) · Zbl 0257.20033 [2] [BL] Brylinski, J.L. Labesse, J.P.: Cohomologies d’intersection et fonctionsL de certaines variétes de Shimura. Ann. Sci. Ec. Norm. Super., IV. Ser.17, 361-412 (1984) · Zbl 0553.12005 [3] [Car] Carayol, H.: Sur les représentationsP-adiques associées aux formes modulaires de Hilbert. Ann. Sci. Ec. Norm. Super., IV. Ser.19, 409-468 (1986) · Zbl 0616.10025 [4] [Cas] Casselman, W.: An assortment of results on representations ofGL 2(k). In: Modular functions of one variable, II. (Lecture Notes in Mathematics, Vol. 349, pp. 1-54). Berlin-Heidelberg-New York: Springer 1973 [5] [De] Demazure, D.: Lectures onp-divisible groups. (Lecture Notes in Mathematics, Vol. 302). Berlin-Heidelberg-New York: Springer 1972 · Zbl 0247.14010 [6] [DR] Deligne, P., Ribet, K.: Values of abelianL-functions at negative integers over totally real fields. Invent. Math.59, 227-286 (1980) · Zbl 0434.12009 [7] [DS] Deligne, P., Serre, J.-P.: Formes modulaires de poids 1. Ann. Sci. Ec. Norm. Super., IV. Ser.7, 507-530 (1974) · Zbl 0321.10026 [8] [G] Greenberg, R.: Onp-adic ArtinL-functions. Nagoya Math. J.89, 77-87 (1983) [9] [HLR] Harder, G., Langlands, R.P., Rapoport, M.: Algebraische Zyklen auf Hilbert-Blumenthal-Flächen. J. Reine Angew. Math.366, 53-120 (1986) · Zbl 0575.14004 [10] [Hi1] 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 [11] [Hi2] Hida, H.: Galois representations intoGL 2(Z p [[x]]) attached to ordinary cusp forms. Invent. Math.85, 546-613 (1986) · Zbl 0612.10021 [12] [Hi3] Hida, H.: On abelian varieties with complex multiplication as factors of the jacobians of Shimura curves. Am. J. Math.103, 727-776 (1981) · Zbl 0477.14024 [13] [Hi4] Hida, H.: Onp-adic Hecke algebras forGL 2 over totally real fields. Preprint [14] [Hi5] Hida, H.: Iwasawa modules attached to congruences of cusp forms. Ann. Sci. Ec. Norm. Supper., IV. Ser.19, 231-273 (1986) · Zbl 0607.10022 [15] [JL] Jacquet, H., Langlands, R.P.: Automorphic forms onGL.(2). (Lecture Notes in Mathematics, Vol. 114). Berlin-Heidelberg-New York: Springer 1970 [16] [JS] Jacquet, H., Shalika, J.: On Euler products and the classification of automorphic forms I and II. Am. J. Math.103, 499-558, 777-815 (1981) · Zbl 0473.12008 [17] [KL] Katz, N.M., Laumon, G.: Transformation de Fourier et majoration de sommes exponentielles. Publ. Math., Inst. Hautes Etud. Sci.62, 145-202 (1986) · Zbl 0603.14015 [18] [MS] Matzushima, Y., Shimura, G.: On the cohomolgy of groups attached to certain vector valued forms on the product of upper half planes. Ann. Math.78, 417-449 (1963) · Zbl 0141.38704 [19] [MW1] Mazur, B., Wiles, A.: Onp-adic analytic families of Galois represenations. Comp. Mech.59, 231-264 (1986) · Zbl 0654.12008 [20] [MW2] Mazur, B., Wiles, A.: Class fields of abelian extensions ofQ. Invent. Math.76, 179-330 (1984) · Zbl 0545.12005 [21] [Ra] Ramakrishnan, D.: Arithmetic of Hilbert-Blumenthal surfaces. Number theory, Proceedings of the Montreal Conference, CMS conference proceedings7, 285-370 (1987) [22] [Ri] Ribet, K.: Congruence relations between modular forms, Proc. International Congress of Mathematicians (1983), pp. 503-514 [23] [RT] Rogawski, J.D., Tunnell, J.B.: On ArtinL-functions associated to Hilbert modular forms of weight one. Invent. Math.74, 1-42 (1983) · Zbl 0523.12009 [24] [Se1] Serre, J.-P.: Abelianl-adic representations and elliptic curves. New York: W.A. Benjamin Inc. 1968 [25] [Se2] Serre, J.-P.: Quelques applications de théoréme de densité de Chebotarev. Publ. Math., Inst. Hautes Etud. Sci.54, 123-202 (1981) · Zbl 0496.12011 [26] [Sh 1] Shimura, G.: The special values of the zeta functions associated with Hilbert modular forms. Duke Math. J.45, 637-679 (1978) · Zbl 0394.10015 [27] [Sh 2] Shimura, G.: The special values of the zeta functions associated with cusp forms. Commun. Pure Appl. Math.29, 783-804 (1976) · Zbl 0348.10015 [28] [Sh3] Shimura, G.: Anl-adic method in the theory of automorphic forms. Unpublished (1968) [29] [Si] Siegel, C.: Über die Fouriersche Koeffizienten von Modulformen. Gött. Nach.3, 15-56 (1970) · Zbl 0225.10031 [30] [T] Tate, J.: Number theoretic background. In: Automorphic forms, representations andL-functions, Proc. Symp. Pure Math., 33, (part 2) 3-26 (1979) [31] [W1] Wiles, A.: Onp-adic representations for totally real fields. Ann. Math.123, 407-456 (1986) · Zbl 0613.12013 [32] [W2] Wiles, A.: The Iwawawa conjecture for totally real fields. (Submitted to Ann. Math.) 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.