E is semistable, p is an odd prime, and Â¯Ï�E,p is irreducible and modular, then (see the proof of Proposition 3.3) and Â¯Ï�E,p satisï¬�es (ii) Â¯Ï�E,p is D 14 of Appendix C of [39]). Therefore by Propositions and (iii) (use Tate curves; see 4.3 and 3.3, Theorem 5.3 implies that the Semistable Modular Lifting Conjecture (Conjecture 2.1) holds for p = 3 and for p = 5. As shown in 2, the Semistable Taniyama-Shimura Conjecture and Fermatâ€™s Last Theorem follow.\n\nmodular for some\n\nD\n\nÂ§\n\nÂ§\n\nTheorem 5.4 (Wiles). Suppose p, k, contains no nontrivial p-th roots of unity. Suppose also that there are an imaginary quadratic ï¬�eld F of discriminant prime to p and a character Ï‡ : Gal( Â¯Q/F ) Ã— such that T is the induced representation IndÏ‡ of GQ is a ( an isomorphism.\n\n, Â¯Ï�, and\n\nare as in\n\n4.2 and\n\nD\n\nO\n\nÂ§\n\nO\n\nâ†’ O\n\n)-lifting of Â¯Ï�. Then Ï• : R\n\n,\n\nD\n\nO\n\nâ†’\n\nCorollary 5.5 (Wiles). Suppose E is an elliptic curve over Q with complex mul- tiplication by an imaginary quadratic ï¬�eld F and p is an odd prime at which E has good reduction. If Eâ€² is an elliptic curve over Q satisfying\n\nEâ€² has good reduction at p and Â¯Ï�Eâ€²,p is isomorphic to Â¯Ï�E,p,\n\nâ€¢\n\nthen Eâ€² is modular.\n\nProof of corollary. Let p be a prime of F containing p, and deï¬�ne = the ring of integers of the completion of F at p,\n\nO â€¢ â€¢ â€¢\n\n/p primes at which E or Eâ€² has bad reduction\n\nk = Î£ = t = ordinary if E has ordinary reduction at p, t = ï¬‚at if E has supersingular reduction at p,\n\n,\n\nO {\n\nO\n\np\n\n,\n\n} âˆª {\n\n}\n\n= (Î£, t).\n\nD\n\nLet\n\nÏ‡ : Gal( Â¯Q/F )\n\nAutO(E[pâˆ�]) âˆ¼=\n\nÃ—\n\nâ†’\n\nO\n\nbe the character giving the action of Gal( Â¯Q/F ) on E[pâˆ�] (where E[pâˆ�] is the group of points of E killed by the endomorphisms of E which lie in some power of p). It is not hard to see that Ï�E,p âŠ— O\n\nis isomorphic to IndÏ‡.\n\n20\n\nK. RUBIN AND A. SILVERBERG\n\nSince E has complex multiplication, it is well
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: E is semistable, p is an odd prime, and Â¯Ï�E,p is irreducible and modular, then (see the proof of Proposition 3.3) and Â¯Ï�E,p satisï¬�es (ii) Â¯Ï�E,p is D 14 of Appendix C of [39]). Therefore by Propositions and (iii) (use Tate curves; see 4.3 and 3.3, Theorem 5.3 implies that the Semistable Modular Lifting Conjecture (Conjecture 2.1) holds for p = 3 and for p = 5. As shown in 2, the Semistable Taniyama-Shimura Conjecture and Fermatâ€™s Last Theorem follow.\n\nmodular for some\n\nD\n\nÂ§\n\nÂ§\n\nTheorem 5.4 (Wiles). Suppose p, k, contains no nontrivial p-th roots of unity. Suppose also that there are an imaginary quadratic ï¬�eld F of discriminant prime to p and a character Ï‡ : Gal( Â¯Q/F ) Ã— such that T is the induced representation IndÏ‡ of GQ is a ( an isomorphism.\n\n, Â¯Ï�, and\n\nare as in\n\n4.2 and\n\nD\n\nO\n\nÂ§\n\nO\n\nâ†’ O\n\n)-lifting of Â¯Ï�. Then Ï• : R\n\n,\n\nD\n\nO\n\nâ†’\n\nCorollary 5.5 (Wiles). Suppose E is an elliptic curve over Q with complex mul- tiplication by an imaginary quadratic ï¬�eld F and p is an odd prime at which E has good reduction. If Eâ€² is an elliptic curve over Q satisfying\n\nEâ€² has good reduction at p and Â¯Ï�Eâ€²,p is isomorphic to Â¯Ï�E,p,\n\nâ€¢\n\nthen Eâ€² is modular.\n\nProof of corollary. Let p be a prime of F containing p, and deï¬�ne = the ring of integers of the completion of F at p,\n\nO â€¢ â€¢ â€¢\n\n/p primes at which E or Eâ€² has bad reduction\n\nk = Î£ = t = ordinary if E has ordinary reduction at p, t = ï¬‚at if E has supersingular reduction at p,\n\n,\n\nO {\n\nO\n\np\n\n,\n\n} âˆª {\n\n}\n\n= (Î£, t).\n\nD\n\nLet\n\nÏ‡ : Gal( Â¯Q/F )\n\nAutO(E[pâˆ�]) âˆ¼=\n\nÃ—\n\nâ†’\n\nO\n\nbe the character giving the action of Gal( Â¯Q/F ) on E[pâˆ�] (where E[pâˆ�] is the group of points of E killed by the endomorphisms of E which lie in some power of p). It is not hard to see that Ï�E,p âŠ— O\n\nis isomorphic to IndÏ‡.\n\n20\n\nK. RUBIN AND A. SILVERBERG\n\nSince E has complex multiplication, it is well