×

zbMATH — the first resource for mathematics

On modular representations of \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\) arising from modular forms. (English) Zbl 0773.11039
In this paper the author proves a conjecture of Serre on the level of an irreducible modular Galois representation \(\rho:\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\to\text{GL}(2,\mathbb F)\), where \(\mathbb F\) is a finite field of odd characteristic \(\ell\). Followed by an idea of G. Frey [Ann. Univ. Sarav., Ser. Math. 1, 1–40 (1986; Zbl 0586.10010)], the main result of this paper has the remarkable application that the Taniyama-Shimura-Weil conjecture (i.e. every elliptic curve over \(\mathbb Q\) is modular) implies Fermat’s Last Theorem.
The representation \(\rho\) is said to be modular of level \(N\) if it arises from a weight-2 newform of level dividing \(N\) and trivial “Nebentypus character”. We say that \(\rho\) is “finite at \(p\)” if there is a finite flat \(\mathbb F\)-vector space scheme \(H\) over \(\mathbb Z_ p\) for which the action of \(\text{Gal}(\overline{\mathbb Q}_ p/\mathbb Q_ p)\) on the \(\mathbb F\)- vector space \(H(\overline{\mathbb Q}_ p)\) gives \(\rho_ p\), where \(\rho_ p\) is the restriction of \(\rho\) to the decomposition group \(\text{Gal}(\overline{\mathbb Q}_ p/\mathbb Q_ p)\) of \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\). If \(\ell\neq p\), this means simply that \(\rho\) is unramified at \(p\). J.-P. Serre conjectured [Contemp. Math. 67, 263–268 (1987; Zbl 0629.14016)] that if \(\rho\) is modular of level \(N\) and if \(\rho\) is finite at a prime \(p\) which divides \(N\) exactly once, then \(\rho\) is also modular of level \(N/p\). Mazur proved this conjecture in the case of \(p\not\equiv 1\pmod\ell\). The main theorem of this paper asserts that Serre’s conjecture is true whenever \(N\) is not divisible by \(\ell\).
Besides Mazur’s techniques, the paper makes use of results of Néron models of Jacobians (due to Raynaud) and of the bad reduction of classical modular curves (Deligne–Rapoport) and Shimura curves (Cherednik–Drinfel’d). Particularly, the author developed a beautiful interchange principle – analogous to the Jacquet-Langlands correspondence – which compares certain data obtained from Shimura curves in characteristic \(p\) to corresponding data obtained from certain modular curves in characteristic \(q\neq p\).

MSC:
11F80 Galois representations
11G18 Arithmetic aspects of modular and Shimura varieties
11G05 Elliptic curves over global fields
11S37 Langlands-Weil conjectures, nonabelian class field theory
14G35 Modular and Shimura varieties
PDF BibTeX Cite
Full Text: DOI EuDML
References:
[1] Atkin, A.O.L., Lehner, J.: Hecke operators onF o(m). Math. Ann.185, 134-160 (1970) · Zbl 0185.15502
[2] Carayol, H.: Sur la mauvaise réduction des courbes de Shimura. Compos. Math.59, 151-230 (1986) · Zbl 0607.14021
[3] Cerednik, I.V.: Uniformization of algebraic curves by discrete arithmetic subgroups of PGL2(k w ) with compact quotients (in Russian). Math. Sb.100, 59-88 (1976). Translation in Math. USSR Sb.29, 55-78 (1976)
[4] Deligne, P., Rapoport, M.: Les schémas de modules de courbes elliptiques. (Lecture Notes in Math. Vol.349, pp. 143-316.) Berlin-Heidelberg-New York: Springer 1973 · Zbl 0281.14010
[5] Deligne, P., Serre, J.-P.: Formes modulaires de poids 1. Ann. Sci. Ec. Norm. Sup., IV. Ser.7, 507-530 (1974). · Zbl 0321.10026
[6] Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Semin. Univ. Hamb.14, 197-272 (1941) · Zbl 0025.02003
[7] Drinfeld, V.G.: Coverings ofp-adic symmetric retions (in Russian). Funkts. Anal. Prilozn10, 29-40 (1976). Translation in Funct. Anal. Appl.10, 107-115 (1976)
[8] Edixhoven, S.J.: L’action de l’algèbre de Hecke sur les groupes de composantes des jacobiennes des courbes modulaires est ?Eisenstein?. Preprint · Zbl 0781.14019
[9] Eichler, M.: Quadratische Formen und Modulfunktionen. Acta Arith.4, 217-239 (1958) · Zbl 0086.06604
[10] Frey, G.: Links between stable elliptic curves and certain diophantine equations. Ann. Univ. Sarav., Ser Math.1, 1-40 (1986) · Zbl 0586.10010
[11] Grothendieck, A.: SGA7 I, exposé IX. (Lecture Notes in Math. Vol.288, pp. 313-523.) Berlin-Heidelberg-New York: Springer 1972
[12] Jacquet, H., Langlands, R.P.: Automorphic forms on GL(2). (Lecture Notes in Math. Vol.114). Berlin-Heidelberg-New York: Springer 1970 · Zbl 0236.12010
[13] Jordan, B., Livné, R.: local diophantine properties of Shimura curves. Math. Ann.270, 235-248 (1985) · Zbl 0548.14010
[14] Jordan, B., Livné, R.: On the Néron model of Jacobians of Shimura curves. Compos. Math.60, 227-236 (1986) · Zbl 0609.14018
[15] Katz, N.M., Mazur, B.: Arithmetic Moduli of Elliptic Curves. Ann. Math. Stud.108 (1985) · Zbl 0576.14026
[16] Kurihara, A.: On some examples of equations defining Shimura curves and the Mumford uniformization. J. Fac., Sci., Univ. Tokyo, Sec. IA25, 277-300 (1979) · Zbl 0428.14012
[17] Langlands, R.P.: Some contemporary problems with origins in the Jugendtraum. Proc. Symp. Pure Math.28, 401-418 (1976) · Zbl 0345.14006
[18] Mazur, B.: Modular curves and the Eisenstein ideal. Publ. Math., Inst. Hautes Etud. Sci.47, 33-186 (1977) · Zbl 0394.14008
[19] Mazur, B.: Rational isogenies of prime degree. Invent. Math.44, 129-162 (1978) · Zbl 0386.14009
[20] Mazur, B.: Letter to J-F. Mestre (16 August 1985)
[21] Mazur, B., Ribet, K.: Two-dimensional represenations in the arithmetic of modular curves. Astérisque (to appear) · Zbl 0780.14015
[22] Milne, J.: Etude d’une classe d’isogénie. In: Variétés de Shimura et FonctionsL, Breen, L., Labesse, J-P. (eds.) Publ. Math. Univ. Paris VII6, 73-81 (1979)
[23] Oda, T.: The first de Rham cohomology group and Dieudonné modules. Ann. Sci. Ec. Norm. Sup. IV, Ser.2, 63-135 (1969) · Zbl 0175.47901
[24] Raynaud, M.: Specialisation du foncteur de Picard. Publ. Math., Inst. Hautes Etud. Sci.38, 27-76 (1970) · Zbl 0207.51602
[25] Raynaud, M.: Schémas en groupes de type (p,...p). Bull. Soc. Math. France102, 241-280 (1974) · Zbl 0325.14020
[26] Ribet, K.: Sur les variétés abéliennes à multiplications réelles. C.R. Acad. Sc. Paris, Sér. A291, 121-123 (1980) · Zbl 0442.14014
[27] Ribet, K.: Modp Hecke operators and congruences between modular forms. Invent. Math.71, 193-205 (1983) · Zbl 0508.10018
[28] Ribet, K.: Congruence relations between modular forms. Proc. Int. Congr. Math. pp. 503-514 (1983)
[29] Ribet, K.: Bimodules and abelian surfaces. Adv. Stud. Pure Math.17, 359-407 (1989) · Zbl 0742.11033
[30] Ribet, K.: On the component groups and the Shimura subgroup ofJ o (N). Sém. Th. Nombres, Université Bordeaux, 1987-88
[31] Ribet, K.: Raising the levels of modular representations, Séminaire de Théorie des Nombres, Paris 1987-88. Progr. Math.81, 259-271 (1990)
[32] Serre, J.-P., Complex multiplication. In: Algebraic Number Theory, Cassels, JWS, Fröhlich, A. (eds.). Washington, DC: Thompson Book Company 1967
[33] Serre, J-P.: Arbres, Amalgames, SL2. Astérisque46 (1977). English translation: Trees. Berlin-Heidelberg-New York: Springer 1980
[34] Serre, J-P.: Lettre à J-F. Mestre (13 août 1985). Contemp. Math.67, 263-268 (1987)
[35] Serre, J-P.: Sur les représentations modulaires de degré 2 de Gal \((\bar Q/Q)\) . Duke Math. J.54, 179-230 (1987) · Zbl 0641.10026
[36] Shimizu, H.: On the zeta functions of quaternion algebras. Ann. Math.81, 166-193 (1965) · Zbl 0201.37903
[37] Shimura, G.: Construction of class fields and zeta functions of algebraic curves. Ann. Math.85, 58-159 (1967) · Zbl 0204.07201
[38] Shimura, G.: On canonical models of arithmetic quotients of bounded symmetric domains. Ann. Math.91, 144-222 (1970) · Zbl 0237.14009
[39] Shimura, G.: Introduction to the Airthmetic Theory of Automorphic Functions. Princeton: Princeton University Press 1971 · Zbl 0221.10029
[40] Tate, J.: Endomorphisms of abelian varieties over finite fields. Invent. Math.2, 134-144 (1966) · Zbl 0147.20303
[41] Vignéras, M-F.: Arithmétíque des Algèbres de Quaternions. (Lecture Notes in Math., Vol.800. Berlin-Heidelberg-New York: Springer 1980
[42] Waterhouse, W.C.: Abelian varieties over finite fields. Ann. Sci. Ec. Norm. Sup. IV. Ser.2, 521-560 (1969) · Zbl 0188.53001
[43] [EGAIV] Grothendieck, A.: Etude locale des schémas et des morphismes de schémas (quatrième partie). Publ. Math, Inst. Hautes Etud. Sci.32, 5-361 (1967)
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.