homomorphism ÏˆÏ� : T\n\nA such that the diagram\n\nO\n\nâ†’ [Ï� T]\n\nâœ²\n\nGL2(T)\n\nGQ\n\nâ��\n\nâ��â��\n\nÏˆÏ� â�„ GL2(A)\n\n[Ï�]\n\nâ��â��â�¥\n\ncommutes.\n\nSince Ï�T is a (\n\n,\n\n)-lifting of Â¯Ï�, by Theorem 4.1 there is a homomorphism\n\nD\n\nO\n\nT\n\nÏ• : R\n\nâ†’\n\nÏ�R. By (T3), Ï•(trace(Ï�R(Frobq))) = Tq for every\n\nsuch that Ï�T is isomorphic to Ï• prime q / âˆˆ\n\nÎ£, so it follows from (T2) that Ï• is surjective.\n\n4.3. Mazur Conjecture, revisited. Conjecture 3.2 can be reformulated in the following way.\n\nConjecture 4.2 (Mazur). Suppose p, k, T is an isomorphism. above map Ï• : R\n\n, Â¯Ï�, and\n\nare as in\n\n4.2. Then the\n\nD\n\nO\n\nÂ§\n\nâ†’\n\nConjecture 4.2 was stated in [23] (Conjecture 18) for\n\nordinary, and Wiles\n\nD\n\nmodiï¬�ed the conjecture to include the ï¬‚at case.\n\nProposition 4.3. Conjecture 4.2 implies Conjecture 3.2.\n\nProof. Suppose Â¯Ï� : GQ -modular, A is D the ring of integers of a ï¬�nite extension of Qp, and Ï� is a type- lifting of Â¯Ï� to A. to be the ring of integers of a suï¬ƒciently large ï¬�nite extension of Qp, and Taking and its residue ï¬�eld, respectively, we may assume that Ï� is extending Ï� and Â¯Ï� to A, with Ï†Ï� a ( as in Theorem 4.1. By (T3) and Theorem 4.1, Ïˆ(Tq) = trace(Ï�(Frobq)) for all but 3.5 of [35], given such a homomorphism Ïˆ (and viewing A as ï¬�nitely many q. By âˆ� n=1 ane2Ï€inz where aq = Ïˆ(Tq) for all but a subring of C), there is an eigenform ï¬�nitely many primes q. Thus Ï� is modular.\n\nGL2(k) is absolutely irreducible and\n\nâ†’\n\nD\n\nO )-lifting of Â¯Ï�. Assuming Conjecture 4.2, let Ïˆ = Ï†Ï� â—¦\n\nO\n\nÏ•âˆ’1 : T\n\n,\n\nD\n\nO\n\nâ†’\n\nÂ§\n\nP\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n15\n\n5. Wilesâ€™ approach to the Mazur Conjecture\n\nIn this section we sketch the major ideas of Wilesâ€™ attack on Conjecture 4.2. The ï¬�rst step (Theorem 5.2), and the key to Wilesâ€™ proof, is to reduce Conjecture 4.2 to a bound on the order of the cotangent space at a prime
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: homomorphism ÏˆÏ� : T\n\nA such that the diagram\n\nO\n\nâ†’ [Ï� T]\n\nâœ²\n\nGL2(T)\n\nGQ\n\nâ��\n\nâ��â��\n\nÏˆÏ� â�„ GL2(A)\n\n[Ï�]\n\nâ��â��â�¥\n\ncommutes.\n\nSince Ï�T is a (\n\n,\n\n)-lifting of Â¯Ï�, by Theorem 4.1 there is a homomorphism\n\nD\n\nO\n\nT\n\nÏ• : R\n\nâ†’\n\nÏ�R. By (T3), Ï•(trace(Ï�R(Frobq))) = Tq for every\n\nsuch that Ï�T is isomorphic to Ï• prime q / âˆˆ\n\nÎ£, so it follows from (T2) that Ï• is surjective.\n\n4.3. Mazur Conjecture, revisited. Conjecture 3.2 can be reformulated in the following way.\n\nConjecture 4.2 (Mazur). Suppose p, k, T is an isomorphism. above map Ï• : R\n\n, Â¯Ï�, and\n\nare as in\n\n4.2. Then the\n\nD\n\nO\n\nÂ§\n\nâ†’\n\nConjecture 4.2 was stated in [23] (Conjecture 18) for\n\nordinary, and Wiles\n\nD\n\nmodiï¬�ed the conjecture to include the ï¬‚at case.\n\nProposition 4.3. Conjecture 4.2 implies Conjecture 3.2.\n\nProof. Suppose Â¯Ï� : GQ -modular, A is D the ring of integers of a ï¬�nite extension of Qp, and Ï� is a type- lifting of Â¯Ï� to A. to be the ring of integers of a suï¬ƒciently large ï¬�nite extension of Qp, and Taking and its residue ï¬�eld, respectively, we may assume that Ï� is extending Ï� and Â¯Ï� to A, with Ï†Ï� a ( as in Theorem 4.1. By (T3) and Theorem 4.1, Ïˆ(Tq) = trace(Ï�(Frobq)) for all but 3.5 of [35], given such a homomorphism Ïˆ (and viewing A as ï¬�nitely many q. By âˆ� n=1 ane2Ï€inz where aq = Ïˆ(Tq) for all but a subring of C), there is an eigenform ï¬�nitely many primes q. Thus Ï� is modular.\n\nGL2(k) is absolutely irreducible and\n\nâ†’\n\nD\n\nO )-lifting of Â¯Ï�. Assuming Conjecture 4.2, let Ïˆ = Ï†Ï� â—¦\n\nO\n\nÏ•âˆ’1 : T\n\n,\n\nD\n\nO\n\nâ†’\n\nÂ§\n\nP\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n15\n\n5. Wilesâ€™ approach to the Mazur Conjecture\n\nIn this section we sketch the major ideas of Wilesâ€™ attack on Conjecture 4.2. The ï¬�rst step (Theorem 5.2), and the key to Wilesâ€™ proof, is to reduce Conjecture 4.2 to a bound on the order of the cotangent space at a prime