Ï„ for complex conjugation. Then Ï„ 2 = 1, and by (6), det(Â¯Ï�E,p(Ï„ )) = 1. Since Â¯Ï�E,p is irreducible and p is odd, a simple linear algebra argument now shows that Â¯Ï�E,p is absolutely irreducible.\n\nâˆ’\n\nSince E satisï¬�es (b) of Conjecture 2.1, Â¯Ï�E,p is modular. Let\n\nÎ£ = t = ordinary if E has ordinary or bad reduction at p, t = ï¬‚at if E has supersingular reduction at p,\n\np\n\nprimes q : E has bad reduction at q\n\n,\n\nâ€¢\n\n{\n\n} âˆª {\n\n}\n\n= (Î£, t).\n\nD\n\nUsing the semistability of E, one can show that Ï�E,p is a type- (by combining results of several people; see [28]) that Â¯Ï�E,p is 3.2 then says Ï�E,p is modular. By Theorem 3.1, E is modular.\n\nlifting of Â¯Ï�E,p and -modular. Conjecture\n\nD\n\nD\n\n4. Mazurâ€™s deformation theory\n\nNext we reformulate Conjecture 3.2 as a conjecture (Conjecture 4.2) that the algebras which parametrize liftings and modular liftings of a given representation are isomorphic. It is this form of Mazurâ€™s conjecture that Wiles attacks directly.\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n13\n\n4.1. The universal deformation algebra R. Fix an odd prime p, a ï¬�nite ï¬�eld k of characteristic p, deformation data representation Â¯Ï� : GQ extension of Qp with residue ï¬�eld k.\n\n, and an absolutely irreducible type-\n\nD\n\nD is the ring of integers of a ï¬�nite\n\nGL2(k). Suppose\n\nâ†’\n\nO\n\nDeï¬�nition. We say Ï� : GQ complete noetherian local commutes\n\n)-lifting of Â¯Ï� if Ï� is type-\n\n, A is a â†’ -algebra with residue ï¬�eld k, and the following diagram\n\nGL2(A) is a (\n\n,\n\nD\n\nO\n\nD\n\nO\n\nGL2(A)\n\nâœŸâœŸâœ¯\n\n[Ï�]\n\nâœŸâœŸ\n\nâ�„ GL2(k)\n\nâœ²\n\nGQ\n\n[ Â¯Ï�]\n\nwhere the vertical map is reduction modulo the maximal ideal of A.\n\nTheorem 4.1 (Mazur-Ramakrishna). With p, k, an that for every ( Ï†Ï� : R\n\nas above, there are D GL2(R) of Â¯Ï�, with the property -algebra homomorphism\n\n, Â¯Ï�, and\n\nO\n\nalgebra R and a (\n\n)-lifting Ï�R : GQ )-lifting Ï� of Â¯Ï� to A there is
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: Ï„ for complex conjugation. Then Ï„ 2 = 1, and by (6), det(Â¯Ï�E,p(Ï„ )) = 1. Since Â¯Ï�E,p is irreducible and p is odd, a simple linear algebra argument now shows that Â¯Ï�E,p is absolutely irreducible.\n\nâˆ’\n\nSince E satisï¬�es (b) of Conjecture 2.1, Â¯Ï�E,p is modular. Let\n\nÎ£ = t = ordinary if E has ordinary or bad reduction at p, t = ï¬‚at if E has supersingular reduction at p,\n\np\n\nprimes q : E has bad reduction at q\n\n,\n\nâ€¢\n\n{\n\n} âˆª {\n\n}\n\n= (Î£, t).\n\nD\n\nUsing the semistability of E, one can show that Ï�E,p is a type- (by combining results of several people; see [28]) that Â¯Ï�E,p is 3.2 then says Ï�E,p is modular. By Theorem 3.1, E is modular.\n\nlifting of Â¯Ï�E,p and -modular. Conjecture\n\nD\n\nD\n\n4. Mazurâ€™s deformation theory\n\nNext we reformulate Conjecture 3.2 as a conjecture (Conjecture 4.2) that the algebras which parametrize liftings and modular liftings of a given representation are isomorphic. It is this form of Mazurâ€™s conjecture that Wiles attacks directly.\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n13\n\n4.1. The universal deformation algebra R. Fix an odd prime p, a ï¬�nite ï¬�eld k of characteristic p, deformation data representation Â¯Ï� : GQ extension of Qp with residue ï¬�eld k.\n\n, and an absolutely irreducible type-\n\nD\n\nD is the ring of integers of a ï¬�nite\n\nGL2(k). Suppose\n\nâ†’\n\nO\n\nDeï¬�nition. We say Ï� : GQ complete noetherian local commutes\n\n)-lifting of Â¯Ï� if Ï� is type-\n\n, A is a â†’ -algebra with residue ï¬�eld k, and the following diagram\n\nGL2(A) is a (\n\n,\n\nD\n\nO\n\nD\n\nO\n\nGL2(A)\n\nâœŸâœŸâœ¯\n\n[Ï�]\n\nâœŸâœŸ\n\nâ�„ GL2(k)\n\nâœ²\n\nGQ\n\n[ Â¯Ï�]\n\nwhere the vertical map is reduction modulo the maximal ideal of A.\n\nTheorem 4.1 (Mazur-Ramakrishna). With p, k, an that for every ( Ï†Ï� : R\n\nas above, there are D GL2(R) of Â¯Ï�, with the property -algebra homomorphism\n\n, Â¯Ï�, and\n\nO\n\nalgebra R and a (\n\n)-lifting Ï�R : GQ )-lifting Ï� of Â¯Ï� to A there is