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On the modularity of elliptic curves over $$\mathbb Q$$: wild 3-adic exercises. (English) Zbl 0982.11033
In this paper the authors complete the work begun by A. Wiles [Ann. Math. (2) 141, 443–551 (1995; Zbl 0823.11029)] and R. Taylor and A. Wiles [Ann. Math. (2) 141, 553–572 (1995; Zbl 0823.11030)] to show that all elliptic curves over $$\mathbb Q$$ are modular. By earlier work of B. Conrad, F. Diamond and R. Taylor [J. Am. Math. Soc. 12, 521–567 (1999; Zbl 0923.11085)], it is sufficient to prove the following result.
Theorem B. If $$\bar{\rho} : \text{Gal}(\bar{\mathbb Q}/{\mathbb Q}) \to \text{GL}_2({\mathbb F}_5)$$ is an irreducible continuous representation with cyclotomic discriminant, then $$\bar{\rho}$$ is modular.
For the proof of Theorem B, the authors divide the representations into six classes according to their 3-adic behaviour. In each case, they find an elliptic curve $$E$$ over $$\mathbb Q$$ with $$\bar{\rho} = \bar{\rho}_{E,5}$$ and a very specific form of the mod-3 representation $$\bar{\rho}_{E,3}$$. By the Langlands-Tunnell theorem, $$\bar{\rho}_{E,3}$$ is modular. Then by techniques à la Wiles and Taylor-Wiles, the authors show that $$\rho_{E,3}$$ is modular as well. Whence $$\bar{\rho} = \bar{\rho}_{E,5}$$ is also modular.
Three of the six cases (3-adic conductor at most $$3^2$$) had been dealt with earlier by F. Diamond [Ann. Math. (2) 144, 137–166 (1996; Zbl 0867.11032)] and Conrad, Diamond and Taylor [loc. cit.]. Quoting from the eminently readable introduction: “This leaves the cases $$f = 27$$, $$81$$, and $$243$$, which are complicated by the fact that $$E$$ now only obtains good reduction over a wild extension of $${\mathbb Q}_3$$. In these cases our argument relies essentially on the particular form we have obtained for $$\bar{\rho}_{E,3}$$ (. . .). We do not believe that our methods for deducing the modularity of $$\rho_{E,3}$$ from that of $$\bar{\rho}_{E,3}$$ would work without this key simplification. It seems to be a piece of undeserved good fortune that for each possibility for $$\bar{\rho}|_{I_3}$$ we can find a choice for $$\bar{\rho}_{E,3}$$ for which our methods work”.

##### MSC:
 11G05 Elliptic curves over global fields 11F80 Galois representations
##### Keywords:
elliptic curve; Galois representation; modularity
Full Text:
##### References:
 [1] Pierre Berthelot, Lawrence Breen, and William Messing, Théorie de Dieudonné cristalline. II, Lecture Notes in Mathematics, vol. 930, Springer-Verlag, Berlin, 1982 (French). · Zbl 0753.14041 [2] Christophe Breuil, Schémas en groupe et modules filtrés, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 2, 93 – 97 (French, with English and French summaries). · Zbl 0924.14025 [3] C. Breuil, Groupes $$p$$-divisibles, groupes finis et modules filtrés, Annals of Math. 152 (2000), 489-549. CMP 2001:06 [4] Henri Carayol, Sur les représentations \?-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 409 – 468 (French). · Zbl 0616.10025 [5] Henri Carayol, Sur les représentations galoisiennes modulo \? attachées aux formes modulaires, Duke Math. J. 59 (1989), no. 3, 785 – 801 (French). · Zbl 0703.11027 [6] Brian Conrad, Ramified deformation problems, Duke Math. J. 97 (1999), no. 3, 439 – 513. · Zbl 0997.11042 [7] Brian Conrad, Fred Diamond, and Richard Taylor, Modularity of certain potentially Barsotti-Tate Galois representations, J. Amer. Math. Soc. 12 (1999), no. 2, 521 – 567. · Zbl 0923.11085 [8] A. J. de Jong, Finite locally free group schemes in characteristic \? and Dieudonné modules, Invent. Math. 114 (1993), no. 1, 89 – 137. · Zbl 0812.14030 [9] P. Deligne, Formes modulaires et représentations $$\ell$$-adiques, in: Lecture Notes in Math. 179, Springer-Verlag, 1971, pp. 139-172. · Zbl 0206.49901 [10] Pierre Deligne and Jean-Pierre Serre, Formes modulaires de poids 1, Ann. Sci. École Norm. Sup. (4) 7 (1974), 507 – 530 (1975) (French). · Zbl 0321.10026 [11] Fred Diamond, The refined conjecture of Serre, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993) Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 22 – 37. · Zbl 0853.11031 [12] Fred Diamond, On deformation rings and Hecke rings, Ann. of Math. (2) 144 (1996), no. 1, 137 – 166. · Zbl 0867.11032 [13] Fred Diamond and Richard Taylor, Lifting modular mod \? representations, Duke Math. J. 74 (1994), no. 2, 253 – 269. · Zbl 0809.11025 [14] Torsten Ekedahl, An effective version of Hilbert’s irreducibility theorem, Séminaire de Théorie des Nombres, Paris 1988 – 1989, Progr. Math., vol. 91, Birkhäuser Boston, Boston, MA, 1990, pp. 241 – 249. · Zbl 0729.12005 [15] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361 (French). · Zbl 0153.22301 [16] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349 – 366 (German). , https://doi.org/10.1007/BF01388432 G. Faltings, Erratum: ”Finiteness theorems for abelian varieties over number fields”, Invent. Math. 75 (1984), no. 2, 381 (German). , https://doi.org/10.1007/BF01388572 G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349 – 366 (German). , https://doi.org/10.1007/BF01388432 G. Faltings, Erratum: ”Finiteness theorems for abelian varieties over number fields”, Invent. Math. 75 (1984), no. 2, 381 (German). · Zbl 0588.14026 [17] Jean-Marc Fontaine and Barry Mazur, Geometric Galois representations, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993) Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 41 – 78. · Zbl 0839.14011 [18] Paul Gérardin, Facteurs locaux des algèbres simples de rang 4. I, Reductive groups and automorphic forms, I (Paris, 1976/1977) Publ. Math. Univ. Paris VII, vol. 1, Univ. Paris VII, Paris, 1978, pp. 37 – 77 (French). [19] C. Khare, A local analysis of congruences in the $$(p,p)$$case: Part II, Invent. Math. 143 (2001), no. 1, 129-155. CMP 2001:06 [20] F. Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, Teubner, 1884. · JFM 16.0061.01 [21] Robert P. Langlands, Base change for \?\?(2), Annals of Mathematics Studies, vol. 96, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. · Zbl 0444.22007 [22] J. Manoharmayum, Pairs of mod 3 and mod 5 representations arising from elliptic curves, Math. Res. Lett. 6 (1999), no. 5-6, 735 – 754. · Zbl 0974.11023 [23] B. Mazur, Number theory as gadfly, Amer. Math. Monthly 98 (1991), no. 7, 593 – 610. · Zbl 0764.11021 [24] Michel Raynaud, Schémas en groupes de type (\?,\ldots ,\?), Bull. Soc. Math. France 102 (1974), 241 – 280 (French). · Zbl 0325.14020 [25] Kenneth A. Ribet, Galois representations attached to eigenforms with Nebentypus, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Springer, Berlin, 1977, pp. 17 – 51. Lecture Notes in Math., Vol. 601. · Zbl 0363.10015 [26] N. I. Shepherd-Barron and R. Taylor, \?\?\?2 and \?\?\?5 icosahedral representations, J. Amer. Math. Soc. 10 (1997), no. 2, 283 – 298. · Zbl 1015.11019 [27] Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. · Zbl 0423.12016 [28] Jean-Pierre Serre, Sur les représentations modulaires de degré 2 de \?\?\?(\?/\?), Duke Math. J. 54 (1987), no. 1, 179 – 230 (French). · Zbl 0641.10026 [29] Goro Shimura, On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields, Nagoya Math. J. 43 (1971), 199 – 208. · Zbl 0225.14015 [30] Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Kanô Memorial Lectures, No. 1. · Zbl 0221.10029 [31] G. Shimura, Response to 1996 Steele Prize, Notices of the AMS 43 (1996), 1344-1347. [32] J. T. Tate, \?-divisible groups, Proc. Conf. Local Fields (Driebergen, 1966) Springer, Berlin, 1967, pp. 158 – 183. [33] Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553 – 572. · Zbl 0823.11030 [34] Jerrold Tunnell, Artin’s conjecture for representations of octahedral type, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 173 – 175. · Zbl 0475.12016 [35] André Weil, Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149 – 156 (German). · Zbl 0158.08601 [36] André Weil, Scientific works. Collected papers. Vol. I (1926 – 1951), Springer-Verlag, New York-Heidelberg, 1979 (French). André Weil, Scientific works. Collected papers. Vol. II (1951 – 1964), Springer-Verlag, New York-Heidelberg, 1979 (French). André Weil, Scientific works. Collected papers. Vol. III (1964 – 1978), Springer-Verlag, New York-Heidelberg, 1979 (French). [37] Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443 – 551. · Zbl 0823.11029
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