Ï�R : GQ )-lifting Ï� of Â¯Ï� to A there is a unique\n\n,\n\nO\n\nD\n\nO\n\nâ†’\n\n,\n\nD\n\nO\n\nO\n\nA such that the diagram\n\nâ†’\n\n[Ï�R]\n\nâœ²\n\nGQ\n\nGL2(R)\n\nâ��\n\nâ��â��\n\nÏ†Ï� â�„ GL2(A)\n\n[Ï�]\n\nâ��â��â�¥\n\ncommutes.\n\nThis theorem was proved by Mazur [21] in the case when\n\nis ordinary and is ï¬‚at. Theorem 4.1 determines R and Ï�R up to\n\nD\n\nby Ramakrishna [25] when isomorphism.\n\nD\n\n4.2. The universal modular deformation algebra T. Fix an odd prime p, a , and an absolutely irreducible ï¬�nite ï¬�eld k of characteristic p, deformation data -modular, and ï¬�x an type- representation Â¯Ï� : GQ eigenform f and a prime ideal Î» of lifting of Â¯Ï�. is the ring of integers of a ï¬�nite extension of Qp with Suppose in addition that residue ï¬�eld k, Of,Î» âŠ† O\n\nD\n\nGL2(k). Assume Â¯Ï� is\n\nD\n\nâ†’\n\nD\n\nOf such that Ï�f,Î» is a type-\n\nD\n\nO , and the diagram\n\nGL2(\n\nOf,Î») â�„ GL2(k)\n\nâœŸâœŸâœŸâœ¯ âœ²\n\n[Ï�f,Î»] âœŸ\n\nGQ\n\n[ Â¯Ï�]\n\ncommutes, where the vertical map is the reduction map.\n\n)-lifting of Â¯Ï�, and Wiles constructs a generalized Hecke algebra T which has the following properties (recall that Hecke algebras T (N ) were deï¬�ned in\n\nUnder these assumptions Ï�f,Î» âŠ— O 1.6).\n\nis a (\n\n,\n\nD\n\nO\n\nÂ§\n\n(T1) T is a complete noetherian local\n\nalgebra with residue ï¬�eld k.\n\nO\n\n14\n\nK. RUBIN AND A. SILVERBERG\n\n(T2) There are an integer N divisible only by primes in Î£ and a homomorphism by the Î£. By abuse of notation\n\nfrom the Hecke algebra T (N ) to T such that T is generated over images of the Hecke operators Tq for primes q / âˆˆ we write Tq also for its image in T.\n\nO\n\n(T3) There is a (\n\n,\n\n)-lifting\n\nD\n\nO\n\nGL2(T)\n\nÏ�T : GQ\n\nâ†’\n\nof Â¯Ï� with the property that trace(Ï�T(Frobq)) = Tq for every prime q / âˆˆ\n\nÎ£. )-lifting of Â¯Ï� to A, then there is a unique\n\n(T4) If Ï� is modular and is a (\n\n,\n\nD\n\nO\n\nalgebra homomorphism ÏˆÏ� : T\n\nA such that the
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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�R : GQ )-lifting Ï� of Â¯Ï� to A there is a unique\n\n,\n\nO\n\nD\n\nO\n\nâ†’\n\n,\n\nD\n\nO\n\nO\n\nA such that the diagram\n\nâ†’\n\n[Ï�R]\n\nâœ²\n\nGQ\n\nGL2(R)\n\nâ��\n\nâ��â��\n\nÏ†Ï� â�„ GL2(A)\n\n[Ï�]\n\nâ��â��â�¥\n\ncommutes.\n\nThis theorem was proved by Mazur [21] in the case when\n\nis ordinary and is ï¬‚at. Theorem 4.1 determines R and Ï�R up to\n\nD\n\nby Ramakrishna [25] when isomorphism.\n\nD\n\n4.2. The universal modular deformation algebra T. Fix an odd prime p, a , and an absolutely irreducible ï¬�nite ï¬�eld k of characteristic p, deformation data -modular, and ï¬�x an type- representation Â¯Ï� : GQ eigenform f and a prime ideal Î» of lifting of Â¯Ï�. is the ring of integers of a ï¬�nite extension of Qp with Suppose in addition that residue ï¬�eld k, Of,Î» âŠ† O\n\nD\n\nGL2(k). Assume Â¯Ï� is\n\nD\n\nâ†’\n\nD\n\nOf such that Ï�f,Î» is a type-\n\nD\n\nO , and the diagram\n\nGL2(\n\nOf,Î») â�„ GL2(k)\n\nâœŸâœŸâœŸâœ¯ âœ²\n\n[Ï�f,Î»] âœŸ\n\nGQ\n\n[ Â¯Ï�]\n\ncommutes, where the vertical map is the reduction map.\n\n)-lifting of Â¯Ï�, and Wiles constructs a generalized Hecke algebra T which has the following properties (recall that Hecke algebras T (N ) were deï¬�ned in\n\nUnder these assumptions Ï�f,Î» âŠ— O 1.6).\n\nis a (\n\n,\n\nD\n\nO\n\nÂ§\n\n(T1) T is a complete noetherian local\n\nalgebra with residue ï¬�eld k.\n\nO\n\n14\n\nK. RUBIN AND A. SILVERBERG\n\n(T2) There are an integer N divisible only by primes in Î£ and a homomorphism by the Î£. By abuse of notation\n\nfrom the Hecke algebra T (N ) to T such that T is generated over images of the Hecke operators Tq for primes q / âˆˆ we write Tq also for its image in T.\n\nO\n\n(T3) There is a (\n\n,\n\n)-lifting\n\nD\n\nO\n\nGL2(T)\n\nÏ�T : GQ\n\nâ†’\n\nof Â¯Ï� with the property that trace(Ï�T(Frobq)) = Tq for every prime q / âˆˆ\n\nÎ£. )-lifting of Â¯Ï� to A, then there is a unique\n\n(T4) If Ï� is modular and is a (\n\n,\n\nD\n\nO\n\nalgebra homomorphism ÏˆÏ� : T\n\nA such that the