On the modular representations of degree two of \(\text{Gal}({\overline {\mathbb Q}}/{\mathbb Q})\). (Sur les représentations modulaires de degré 2 de \(\text{Gal}({\overline {\mathbb Q}}/{\mathbb Q})\).) (French) Zbl 0641.10026

Serre, Jean-Pierre

Duke Math. J. 54, 179-230 (1987).

In the present paper the author precises a very interesting conjecture stated in 1973: If \(\rho\) is an irreducible odd representation of \(G_{{\mathbb Q}}\) to \(\mathrm{GL}(2,{\overline {\mathbb F}}_ p)\) then it should be modular in the following sense: There exists a parabolic cusp form \(f\) of level \(N\), weight \(k\) and character \(\varepsilon\) with coefficients in \({\overline {\mathbb F}}_ p\) [cf. N. M. Katz, \(p\)-adic properties of modular schemes and modular forms, Lect. Notes Math. 350, 69–190 (1973; Zbl 0271.10033)] such that the trace of the image of Frobenius elements \(\pi_{\ell}\) of primes \(\ell \nmid N\) is equal to the \(\ell\)th Fourier coefficient of \(f\) and \(\det\;\rho\) \(=\varepsilon \cdot \chi^{k-1}\) (\(\chi\) cyclotomic character mod \(p\)).
Moreover there is a precise recipe how to determine \(N\), \(k\) and \(\varepsilon\): \(N\) is the Artin conductor of \(\rho\) away from \(p\), and \(k\) is determined by the restriction of \(\rho\) to an inertia group \(I_ p\) at \(p\). For instance: \(k\) is equal to 2 iff \(\det\;\rho| I_ p=\chi\) and the group scheme corresponding to \(\rho | G_{{\mathbb Q}_ p}\) has an extension to a flat finite group scheme over \({\mathbb Z}_ p\) (i.e. \(\rho\) is finite at \(p\)).
This conjecture has very strong and interesting consequences which can be found in § 4. For example: Take \(E/{\mathbb Q}\) as elliptic curve which is semi-stable at \(p\), and take \(\rho =\rho_{E_ p}\) as the representation induced by the action of \(G_{{\mathbb Q}}\) on the points of order \(p\) of \(E\). \(\rho_{E_ p}\) is finite at \(p\) (and hence \(k=2)\) iff \(\mathrm{Min}\{0,v_ p(j_ E)\}\equiv 0 \pmod p.\) For given \(E\) this is so for infinitely many \(p\), and so one gets surprisingly that \(E\) is a quotient of the Jacobian of \(X_ 0(N_ E)\) with \(N_ E\) the conductor of \(E\). Hence Serre’s conjecture implies Taniyama’s conjecture claiming that every elliptic curve over \({\mathbb Q}\) is modular. (For an extension to abelian varieties with real multiplications cf. 4.7.)
On the other side a beautiful result of K. Ribet [On modular representations of \(\text{Gal}({\overline {\mathbb Q}}/{\mathbb Q})\) arising from modular forms; Preprint # 06420-87; Math. Res. Inst. Berkeley, CA (1987)] states that if one assumes that \(E/{\mathbb Q}\) is modular and some reasonable conditions are satisfied then Serre’s conjecture is true for \(\rho_{E_ p}\). This can be applied to Fermat’s Last Theorem: A nontrivial solution of \(Z^p_ 1-Z^p_ 2=Z^p_ 3\) \((p\geq 3)\) can be used to define an elliptic curve over \({\mathbb Q}\) which, due to Ribet’s theorem, cannot be modular [cf. the reviewer, Links between stable elliptic curves and certain Diophantine equations, Ann. Univ. Sarav., Ser. Math. 1, No. 1 (1986; Zbl 0586.10010)]. Hence Taniyama’s conjecture (and so Serre’s conjecture) implies Fermat’s Last Theorem.
In the last section of the paper one finds interesting examples (for \(p=2, 3, 7\)) for which Serre’s conjecture is verified (at least partly) mostly by using computer programs implemented by J.-F. Mestre.

Reviewer: G. Frey

MSC:

11F80	Galois representations
11F33	Congruences for modular and \(p\)-adic modular forms
11G05	Elliptic curves over global fields
11R32	Galois theory
11R39	Langlands-Weil conjectures, nonabelian class field theory

Keywords:

modular representation of \(G_{{\mathbb Q}}\); modular forms mod \(p\); modular elliptic curves; group schemes of type \((p,p)\); Taniyama’s conjecture; Serre’s conjecture; Fermat’s Last Theorem

Citations:

Zbl 0271.10033; Zbl 0586.10010

References:

[1]	E. Artin, Zur Theorie der \(L\)-Reihen mit allgemeinen Gruppencharakteren , Hamb. Abh. 8 (1930), 292-306, ( = coll. P., 165-179). · JFM 56.0173.02
[2]	A. Ash and G. Stevens, Cohomology of arithmetic groups and congruences between systems of Hecke eigenvalues , J. Crelle Math. 365 (1986), 192-220. · Zbl 0596.10026
[3]	A. O. L. Atkin and W. Li, Twists of newforms and pseudo-eigenvalues of \(W\)-operators , Invent. Math. 48 (1978), no. 3, 221-243. · Zbl 0369.10016
[4]	B. J. Birch and W. Kuyk, Modular Forms of One Variable, IV , Lect. Notes in Math., vol. 476, Springer-Verlag, 1975. · Zbl 0315.14014
[5]	S. Bloch, Algebraic cycles and values of \(L\)-functions. II , Duke Math. J. 52 (1985), no. 2, 379-397. · Zbl 0628.14006
[6]	A. Brumer and K. Kramer, The rank of elliptic curves , Duke Math. J. 44 (1977), no. 4, 715-743. · Zbl 0376.14011
[7]	J. P. Buhler, Icosahedral Galois Representations , Lecture Notes in Mathematics, vol. 654, Springer-Verlag, Berlin, 1978. · Zbl 0374.12002
[8]	H. Carayol, Sur les représentations \(l\)-adiques associées aux formes modulaires de Hilbert , Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 409-468. · Zbl 0616.10025
[9]	P. Deligne, Les constantes des équations fonctionnelles des fonctions \(L\) , Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, 501-597. Lecture Notes in Math., Vol. 349. · Zbl 0271.14011
[10]	P. Deligne, Valeurs de fonctions \(L\) et périodes d’intégrales , Automorphic forms, representations and \(L\)-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 313-346. · Zbl 0449.10022
[11]	P. Deligne and J.-P. Serre, Formes modulaires de poids \(1\) , Ann. Sci. École Norm. Sup. (4) 7 (1974), 507-530 (1975). · Zbl 0321.10026
[12]	P. Dénes, Über die Diophantische Gleichung \(x^ l+y^ l=cz^ l\) , Acta Math. 88 (1952), 241-251. · Zbl 0048.27503
[13]	1 G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern , Invent. Math. 73 (1983), no. 3, 349-366. · Zbl 0588.14026
[14]	2 G. Faltings, Erratum: “Finiteness theorems for abelian varieties over number fields” , Invent. Math. 75 (1984), no. 2, 381. · Zbl 0588.14026
[15]	G. Faltings, G. Wüstholz, et al., Rational points , Aspects of Mathematics, E6, Friedr. Vieweg & Sohn, Braunschweig, 1984. · Zbl 0588.14027
[16]	J.-M. Fontaine, Il n’y a pas de variété abélienne sur \(\mathbf Z\) , Invent. Math. 81 (1985), no. 3, 515-538. · Zbl 0612.14043
[17]	G. Frey, Rationale Punkte auf Fermatkurven und getwisteten Modulkurven , J. Crelle Math. 331 (1982), 185-191. · Zbl 0474.14011
[18]	G. Frey, Links between stable elliptic curves and certain Diophantine equations , Ann. Univ. Sarav. Ser. Math. 1 (1986), no. 1, iv+40. · Zbl 0586.10010
[19]	A. Grothendieck, Groupes de monodromie en géométrie algébrique. I , Lecture Notes in Mathematics, vol. 288, Springer-Verlag, Berlin, 1972. · Zbl 0237.00013
[20]	Y. Hellegouarch, Courbes elliptiques et équation de Fermat , Thèse, Besançon, 1972.
[21]	A. Hurwitz, Über endliche Gruppen linearer Substitutionen, welche in der Theorie der elliptischen Transzendenten aufreten , Math. Ann. 27 (1886), 183-233, = Math. W. XI. · JFM 19.0472.01
[22]	N. Jochnowitz, A study of the local components of the Hecke algebra mod \(l\) , Trans. Amer. Math. Soc. 270 (1982), no. 1, 253-267. · Zbl 0536.10021
[23]	N. Jochnowitz, Congruences between systems of eigenvalues of modular forms , Trans. Amer. Math. Soc. 270 (1982), no. 1, 269-285. JSTOR: · Zbl 0536.10022
[24]	N. M. Katz, \(p\)-adic properties of modular schemes and modular forms , Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, 69-190. Lecture Notes in Mathematics, Vol. 350. · Zbl 0271.10033
[25]	N. M. Katz, A result on modular forms in characteristic \(p\) , Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Springer, Berlin, 1977, 53-61. Lecture Notes in Math., Vol. 601. · Zbl 0392.10026
[26]	S. LaMacchia, Polynomials with Galois group \(\mathrm PSL(2,\,7)\) , Comm. in Algebra 8 (1980), no. 10, 983-992. · Zbl 0436.12005
[27]	R. P. Langlands, Base change for \(\mathrm GL(2)\) , Annals of Mathematics Studies, vol. 96, Princeton Univ. Press, Princeton, N.J., 1980. · Zbl 0444.22007
[28]	W. Li, Newforms and functional equations , Math. Ann. 212 (1975), 285-315. · Zbl 0278.10026
[29]	B. Mazur, Modular curves and the Eisenstein ideal , Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 33-186 (1978). · Zbl 0278.10026
[30]	B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld) , Invent. Math. 44 (1978), no. 2, 129-162. · Zbl 0386.14009
[31]	J.-F. Mestre, Courbes hyperelliptiques à multiplications réelles , Séminaire de Théorie des Nombres, 1987-1988 (Talence, 1987-1988), Univ. Bordeaux I, Talence, 1988, Exp. No. 34, 6. · Zbl 0688.14035
[32]	J.-F. Mestre, La méthode des graphes. Examples et applications , Taniguchi Symp., Kyoto, 1986, à paraître. · Zbl 0621.14021
[33]	I. Miyawaki, Elliptic curves of prime power conductor with \(\mathbf Q\)-rational points of finite order , Osaka J. Math. 10 (1973), 309-323. · Zbl 0269.14012
[34]	O. Neumann, Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten. II , Math. Nachr. 56 (1973), 269-280. · Zbl 0277.14011
[35]	J. Tate and F. Oort, Group schemes of prime order , Ann. Sci. École Norm. Sup. (4) 3 (1970), 1-21. · Zbl 0195.50801
[36]	M. Raynaud, Schémas en groupes de type \((p,\dots, p)\) , Bull. Soc. Math. France 102 (1974), 241-280. · Zbl 0325.14020
[37]	K. A. Ribet, Galois action on division points of Abelian varieties with real multiplications , Amer. J. Math. 98 (1976), no. 3, 751-804. JSTOR: · Zbl 0348.14022
[38]	C. Schoen, On the geometry of a special determinantal hypersurface associated to the Mumford-Horrocks vector bundle , J. Crelle. Math. 364 (1986), 85-111. · Zbl 0568.14022
[39]	J.-P. Serre, Corps Locaux , 3ème édition ed., Hermann, Paris, 1968. · Zbl 0423.12017
[40]	J.-P. Serre, Représentations linéaires des groupes finis , 3ème édition ed., Hermann, Paris, 1978. · Zbl 0407.20003
[41]	J.-P. Serre, Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures) , 1969/1970, Sém. Delange-Pisot-Poitou, exposé 19 (= Oe. 87). · Zbl 0214.48403
[42]	J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques , Invent. Math. 15 (1972), no. 4, 259-331. · Zbl 0235.14012
[43]	J.-P. Serre, Congruences et formes modulaires [d’après H. P. F. Swinnerton-Dyer] , Séminaire Bourbaki, 24e année (1971/1972), Exp. No. 416, Springer, Berlin, 1973, 319-338. Lecture Notes in Math., Vol. 317. · Zbl 0276.14013
[44]	J.-P. Serre, Formes modulaires et fonctions zêta \(p\)-adiques , Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), Springer, Berlin, 1973, 191-268. Lecture Notes in Math., Vol. 350. · Zbl 0277.12014
[45]	J.-P. Serre, Valeurs propres des opérateurs de Hecke modulo \(l\) , Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, 1974), Soc. Math. France, Paris, 1975, 109-117. Astérisque, Nos. 24-25. · Zbl 0305.10021
[46]	J.-P. Serre, Modular forms of weight one and Galois representations , Algebraic number fields: \(L\)-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) ed. A. Fröhlich, Academic Press, London, 1977, pp. 193-268. · Zbl 0366.10022
[47]	J.-P. Serre, L’invariant de Witt de la forme \(\mathrm Tr(x^ 2)\) , Comment. Math. Helv. 59 (1984), no. 4, 651-676. · Zbl 0565.12014
[48]	J.-P. Serre, Résumé des cours de 1984-1985 , Annuaire du Collège de France (1985), 85-90.
[49]	J.-P. Serre, Lettre à J.-F. Mestre , Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) ed. K. Ribet, Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 263-268. · Zbl 0629.14016
[50]	J.-P. Serre and J. Tate, Good reduction of abelian varieties , Ann. of Math. (2) 88 (1968), 492-517. JSTOR: · Zbl 0172.46101
[51]	B. Setzer, Elliptic curves of prime conductor , J. London Math. Soc. (2) 10 (1975), 367-378. · Zbl 0324.14005
[52]	G. Shimura, Introduction to the arithmetic theory of automorphic functions , no. 11, Publ. Math. Soc. Japan, Princeton Univ. Press, 1971. · Zbl 0221.10029
[53]	G. Shimura, Class fields over real quadratic fields and Hecke operators , Ann. of Math. (2) 95 (1972), 130-190. JSTOR: · Zbl 0255.10032
[54]	J. Tunnell, Artin’s conjecture for representations of octahedral type , Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 173-175. · Zbl 0475.12016
[55]	J. Vélu, Courbes modulaires et courbes de Fermat , Séminaire de Théorie des Nombres, 1975-1976 (Univ. Bordeaux I, Talence), Exp. No. 16, Lab. Théorie des Nombres, Centre Nat. Recherche Sci., Talence, 1976, p. 10. · Zbl 0363.14007
[56]	A. Weil, Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen , Math. Ann. 168 (1967), 149-156. · Zbl 0158.08601