On ordinary \(\lambda\)-adic representations associated to modular forms. (English) Zbl 0664.10013

Wiles, A.

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

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

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.)