to GL2(B) will also be\n\nâ†’\n\ndenoted Ï†. If Ï� : GQ Aâ€² for the composition of Ï� with the inclusion of GL2(A) in GL2(Aâ€²).\n\nGL2(A) is a representation and Aâ€² is a ring containing A, we write\n\nâ†’\n\nÏ�\n\nâŠ—\n\nDeï¬�nition. If Â¯Ï� : GQ Ï� : GQ Zp-algebra and there exists a homomorphism Î¹ : A\n\nGL2(k) is a representation, we say that a representation GL2(A) is a lifting of Â¯Ï� (to A) if A is a complete noetherian local\n\nâ†’\n\nâ†’\n\nÂ¯k such that the diagram\n\nâ†’ GL2(A)\n\nâœŸâœŸâœ¯\n\n[Ï�]\n\nâœŸâœŸ\n\nÎ¹ â�„ GL2(Â¯k)\n\nâœ²\n\nGQ\n\n[ Â¯Ï� âŠ— Â¯k]\n\nÂ¯k].\n\ncommutes, in the sense that [Î¹\n\nÏ�] = [Â¯Ï�\n\n\n\nâŠ—\n\nExamples. (i) If E is an elliptic curve then Ï�E,p is a lifting of Â¯Ï�E,p.\n\n(ii) If E is an elliptic curve, p is a prime, and hypotheses (a) and (b) of Conjecture\n\n2.1 hold with an eigenform f and prime ideal Î», then Ï�f,Î» is a lifting of Â¯Ï�E,p.\n\n3.4. Deformation data. We will be interested not in all liftings of a given Â¯Ï�, but rather in those satisfying various restrictions. See Appendix A for the deï¬�nition of GQ associated to primes q. We say that a representation Ï� the inertia groups Iq âŠ‚ of GQ is unramiï¬�ed at a prime q if Ï�(Iq) = 1. If Î£ is a set of primes, we say Ï� is unramiï¬�ed outside of Î£ if Ï� is unramiï¬�ed at every q / âˆˆ\n\nÎ£.\n\nDeï¬�nition. By deformation data we mean a pair\n\n= (Î£, t)\n\nD where Î£ is a ï¬�nite set of primes and t is one of the words ordinary or ï¬‚at.\n\nZÃ—\n\nAÃ— be the composition of the\n\nIf A is a Zp-algebra, let ÎµA : GQ\n\np â†’\n\nâ†’\n\ncyclotomic character Îµp with the structure map.\n\nDeï¬�nition. Given deformation data type- outside of Î£, and Ï� is t at p (where t\n\nGL2(A) is if A is a complete noetherian local Zp-algebra, det(Ï�) = ÎµA, Ï� is unramiï¬�ed\n\n, a representation Ï� : GQ\n\nD\n\nâ†’\n\nD\n\nordinary, ï¬‚at }\n\n; see Appendix C).\n\nâˆˆ {\n\nDeï¬�nition. A representation Â¯Ï� : GQ eigenform f and a prime ideal Î» of\n\nmodular if
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: to GL2(B) will also be\n\nâ†’\n\ndenoted Ï†. If Ï� : GQ Aâ€² for the composition of Ï� with the inclusion of GL2(A) in GL2(Aâ€²).\n\nGL2(A) is a representation and Aâ€² is a ring containing A, we write\n\nâ†’\n\nÏ�\n\nâŠ—\n\nDeï¬�nition. If Â¯Ï� : GQ Ï� : GQ Zp-algebra and there exists a homomorphism Î¹ : A\n\nGL2(k) is a representation, we say that a representation GL2(A) is a lifting of Â¯Ï� (to A) if A is a complete noetherian local\n\nâ†’\n\nâ†’\n\nÂ¯k such that the diagram\n\nâ†’ GL2(A)\n\nâœŸâœŸâœ¯\n\n[Ï�]\n\nâœŸâœŸ\n\nÎ¹ â�„ GL2(Â¯k)\n\nâœ²\n\nGQ\n\n[ Â¯Ï� âŠ— Â¯k]\n\nÂ¯k].\n\ncommutes, in the sense that [Î¹\n\nÏ�] = [Â¯Ï�\n\n\n\nâŠ—\n\nExamples. (i) If E is an elliptic curve then Ï�E,p is a lifting of Â¯Ï�E,p.\n\n(ii) If E is an elliptic curve, p is a prime, and hypotheses (a) and (b) of Conjecture\n\n2.1 hold with an eigenform f and prime ideal Î», then Ï�f,Î» is a lifting of Â¯Ï�E,p.\n\n3.4. Deformation data. We will be interested not in all liftings of a given Â¯Ï�, but rather in those satisfying various restrictions. See Appendix A for the deï¬�nition of GQ associated to primes q. We say that a representation Ï� the inertia groups Iq âŠ‚ of GQ is unramiï¬�ed at a prime q if Ï�(Iq) = 1. If Î£ is a set of primes, we say Ï� is unramiï¬�ed outside of Î£ if Ï� is unramiï¬�ed at every q / âˆˆ\n\nÎ£.\n\nDeï¬�nition. By deformation data we mean a pair\n\n= (Î£, t)\n\nD where Î£ is a ï¬�nite set of primes and t is one of the words ordinary or ï¬‚at.\n\nZÃ—\n\nAÃ— be the composition of the\n\nIf A is a Zp-algebra, let ÎµA : GQ\n\np â†’\n\nâ†’\n\ncyclotomic character Îµp with the structure map.\n\nDeï¬�nition. Given deformation data type- outside of Î£, and Ï� is t at p (where t\n\nGL2(A) is if A is a complete noetherian local Zp-algebra, det(Ï�) = ÎµA, Ï� is unramiï¬�ed\n\n, a representation Ï� : GQ\n\nD\n\nâ†’\n\nD\n\nordinary, ï¬‚at }\n\n; see Appendix C).\n\nâˆˆ {\n\nDeï¬�nition. A representation Â¯Ï� : GQ eigenform f and a prime ideal Î» of\n\nmodular if