at p if Ï� is not ordinary, and for every ideal a of ï¬�nite index in A, the (cid:0) |Dp modulo a is the representation associated to the Â¯Qp-points reduction of Ï� of a ï¬�nite ï¬‚at group scheme over Zp.\n\n\n\nâˆ— âˆ— 0 Ï‡\n\nwhere Ï‡ is unramiï¬�ed and the * are functions from Dp to A;\n\n(cid:1)\n\n\n\nAppendix D. Selmer groups\n\nWith notation as in\n\n5 (see especially Â§\n\n5.2), deï¬�ne\n\nÂ§\n\n[Ç«]/(Ç«2, mn)\n\nOn =\n\nO\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n23\n\nwhere Ç« is an indeterminate. Then v\n\n1 + Ç«v deï¬�nes an isomorphism\n\n7â†’ On) : Î´ GL2(\n\nâˆ¼ âˆˆ â†’ { HomO(pR/p2 R,\n\n(16)\n\n1 (mod Ç«) } /mn) there is a unique -algebra homomorphism â†’ On whose restriction to pR is Ç«Î±. Composing with the representation Ï�R On. (In particular Ï�0 )-lifting obtained when Î± = 0.) Deï¬�ne a one-cocycle cÎ± on GQ\n\nÎ´\n\n.\n\nVn\n\nâ‰¡\n\nFor every Î±\n\nO\n\nO\n\nâˆˆ\n\nÏˆÎ± : R of Theorem 4.1 gives a ( denotes the ( by\n\n,\n\n)-lifting Ï�Î± = ÏˆÎ± â—¦\n\nÏ�R of Â¯Ï� to\n\nD\n\nO\n\n,\n\nD\n\nO\n\ncÎ±(g) = Ï�Î±(g)Ï�0(g)âˆ’1.\n\nH 1(Q, Vn). This deï¬�nes a\n\nSince Ï�Î± â‰¡ homomorphism\n\nÏ�0 (mod Ç«), using (16) we can view cÎ± âˆˆ\n\ns : HomO(pR/p2 R,\n\n/mn)\n\nH 1(Q, Vn),\n\nO and it is not diï¬ƒcult to see that s is injective. The fact that Ï�0 and Ï�Î± are type- D gives information about the restrictions resq(cÎ±) for various primes q, and using this H 1(Q, Vn) and veriï¬�es that s information Wiles deï¬�nes a Selmer group SD(Vn) is an isomorphism onto SD(Vn).\n\nâ†’\n\nâŠ‚\n\nReferences\n\n[1] B. Birch and W. Kuyk, eds., Modular functions of one variable. IV, Lecture Notes in Math.,\n\nvol. 476, Springer-Verlag, New York, 1975, pp. 74â€“144.\n\n[2] J. Buhler, R. Crandall, R. Ernvall, and T. MetsÂ¨ankylÂ¨a, Irregular primes and cyclotomic\n\ninvariants to four million, Math. Comp. 61 (1993), 151â€“153.\n\n[3] J. W. S. Cassels and A. Frohlich, Algebraic number theory, Academic Press, London, 1967. [4] P. Deligne and J.-P. Serre,
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: at p if Ï� is not ordinary, and for every ideal a of ï¬�nite index in A, the (cid:0) |Dp modulo a is the representation associated to the Â¯Qp-points reduction of Ï� of a ï¬�nite ï¬‚at group scheme over Zp.\n\n\n\nâˆ— âˆ— 0 Ï‡\n\nwhere Ï‡ is unramiï¬�ed and the * are functions from Dp to A;\n\n(cid:1)\n\n\n\nAppendix D. Selmer groups\n\nWith notation as in\n\n5 (see especially Â§\n\n5.2), deï¬�ne\n\nÂ§\n\n[Ç«]/(Ç«2, mn)\n\nOn =\n\nO\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n23\n\nwhere Ç« is an indeterminate. Then v\n\n1 + Ç«v deï¬�nes an isomorphism\n\n7â†’ On) : Î´ GL2(\n\nâˆ¼ âˆˆ â†’ { HomO(pR/p2 R,\n\n(16)\n\n1 (mod Ç«) } /mn) there is a unique -algebra homomorphism â†’ On whose restriction to pR is Ç«Î±. Composing with the representation Ï�R On. (In particular Ï�0 )-lifting obtained when Î± = 0.) Deï¬�ne a one-cocycle cÎ± on GQ\n\nÎ´\n\n.\n\nVn\n\nâ‰¡\n\nFor every Î±\n\nO\n\nO\n\nâˆˆ\n\nÏˆÎ± : R of Theorem 4.1 gives a ( denotes the ( by\n\n,\n\n)-lifting Ï�Î± = ÏˆÎ± â—¦\n\nÏ�R of Â¯Ï� to\n\nD\n\nO\n\n,\n\nD\n\nO\n\ncÎ±(g) = Ï�Î±(g)Ï�0(g)âˆ’1.\n\nH 1(Q, Vn). This deï¬�nes a\n\nSince Ï�Î± â‰¡ homomorphism\n\nÏ�0 (mod Ç«), using (16) we can view cÎ± âˆˆ\n\ns : HomO(pR/p2 R,\n\n/mn)\n\nH 1(Q, Vn),\n\nO and it is not diï¬ƒcult to see that s is injective. The fact that Ï�0 and Ï�Î± are type- D gives information about the restrictions resq(cÎ±) for various primes q, and using this H 1(Q, Vn) and veriï¬�es that s information Wiles deï¬�nes a Selmer group SD(Vn) is an isomorphism onto SD(Vn).\n\nâ†’\n\nâŠ‚\n\nReferences\n\n[1] B. Birch and W. Kuyk, eds., Modular functions of one variable. IV, Lecture Notes in Math.,\n\nvol. 476, Springer-Verlag, New York, 1975, pp. 74â€“144.\n\n[2] J. Buhler, R. Crandall, R. Ernvall, and T. MetsÂ¨ankylÂ¨a, Irregular primes and cyclotomic\n\ninvariants to four million, Math. Comp. 61 (1993), 151â€“153.\n\n[3] J. W. S. Cassels and A. Frohlich, Algebraic number theory, Academic Press, London, 1967. [4] P. Deligne and J.-P. Serre,