: GQ\n\nâ†’\n\nsuch that det(Ï�E,p) = Îµp and for all but ï¬�nitely many primes q,\n\n#(E(Fq)).\n\n(11)\n\ntrace(Ï�E,p(Frobq)) = q + 1\n\nâˆ’\n\nComposing Ï�E,p with the reduction map from Zp to Fp gives Â¯Ï�E,p of\n\n2.1. Â§\n\n3.2. Modular representations. If f is an eigenform and Î» is a prime ideal of Of at Î». Of , let\n\nOf,Î» denote the completion of\n\nDeï¬�nition. If A is a ring, a representation Ï� : GQ if there are an eigenform f (z) = homomorphism Î¹ :\n\nGL2(A) is called modular âˆ� n=1 ane2Ï€inz, a ring Aâ€² containing A, and a\n\nâ†’\n\nAâ€² such that for all but ï¬�nitely many primes q,\n\nOf â†’\n\nP\n\ntrace(Ï�(Frobq)) = Î¹(aq).\n\nâˆ� n=1 ane2Ï€inz and a prime ideal Î» of\n\nExamples. (i) Given an eigenform f (z) = Of , Eichler and Shimura (see\n\n7.6 of [35]) constructed a representation\n\nÂ§\n\nP\n\nÏ�f,Î» : GQ\n\nGL2(\n\nOf,Î»)\n\nâ†’\n\nZ = pZ) and for all but ï¬�nitely many primes q,\n\nsuch that det(Ï�f,Î») = Îµp (where Î»\n\nâˆ©\n\n(12)\n\ntrace(Ï�f,Î»(Frobq)) = aq.\n\nThus Ï�f,Î» is modular with Î¹ taken to be the inclusion of\n\nOf in\n\nOf,Î».\n\n(ii) Suppose p is a prime and E is an elliptic curve over Q. If E is modular, then Ï�E,p and Â¯Ï�E,p are modular by (11), (7), and (5). Conversely, if Ï�E,p is modular, then it follows from (11) that E is modular. This proves the following.\n\nTheorem 3.1. Suppose E is an elliptic curve over Q. Then\n\nE is modular\n\nÏ�E,p is modular for every p\n\nÏ�E,p is modular for one p.\n\nâ‡”\n\nâ‡”\n\nRemark . In this language, the Semistable Modular Lifting Conjecture says that if p is an odd prime, E is a semistable elliptic curve over Q, and Â¯Ï�E,p is modular and irreducible, then Ï�E,p is modular.\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n11\n\n3.3. Liftings of Galois representations. Fix a prime p and a ï¬�nite ï¬�eld k of characteristic p. Recall that Â¯k denotes an algebraic closure of k.\n\nGiven a map Ï† : A\n\nB, the induced map from GL2(A) to GL2(B) will also be\n\nâ†’\n\ndenoted Ï†.
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: : GQ\n\nâ†’\n\nsuch that det(Ï�E,p) = Îµp and for all but ï¬�nitely many primes q,\n\n#(E(Fq)).\n\n(11)\n\ntrace(Ï�E,p(Frobq)) = q + 1\n\nâˆ’\n\nComposing Ï�E,p with the reduction map from Zp to Fp gives Â¯Ï�E,p of\n\n2.1. Â§\n\n3.2. Modular representations. If f is an eigenform and Î» is a prime ideal of Of at Î». Of , let\n\nOf,Î» denote the completion of\n\nDeï¬�nition. If A is a ring, a representation Ï� : GQ if there are an eigenform f (z) = homomorphism Î¹ :\n\nGL2(A) is called modular âˆ� n=1 ane2Ï€inz, a ring Aâ€² containing A, and a\n\nâ†’\n\nAâ€² such that for all but ï¬�nitely many primes q,\n\nOf â†’\n\nP\n\ntrace(Ï�(Frobq)) = Î¹(aq).\n\nâˆ� n=1 ane2Ï€inz and a prime ideal Î» of\n\nExamples. (i) Given an eigenform f (z) = Of , Eichler and Shimura (see\n\n7.6 of [35]) constructed a representation\n\nÂ§\n\nP\n\nÏ�f,Î» : GQ\n\nGL2(\n\nOf,Î»)\n\nâ†’\n\nZ = pZ) and for all but ï¬�nitely many primes q,\n\nsuch that det(Ï�f,Î») = Îµp (where Î»\n\nâˆ©\n\n(12)\n\ntrace(Ï�f,Î»(Frobq)) = aq.\n\nThus Ï�f,Î» is modular with Î¹ taken to be the inclusion of\n\nOf in\n\nOf,Î».\n\n(ii) Suppose p is a prime and E is an elliptic curve over Q. If E is modular, then Ï�E,p and Â¯Ï�E,p are modular by (11), (7), and (5). Conversely, if Ï�E,p is modular, then it follows from (11) that E is modular. This proves the following.\n\nTheorem 3.1. Suppose E is an elliptic curve over Q. Then\n\nE is modular\n\nÏ�E,p is modular for every p\n\nÏ�E,p is modular for one p.\n\nâ‡”\n\nâ‡”\n\nRemark . In this language, the Semistable Modular Lifting Conjecture says that if p is an odd prime, E is a semistable elliptic curve over Q, and Â¯Ï�E,p is modular and irreducible, then Ï�E,p is modular.\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n11\n\n3.3. Liftings of Galois representations. Fix a prime p and a ï¬�nite ï¬�eld k of characteristic p. Recall that Â¯k denotes an algebraic closure of k.\n\nGiven a map Ï† : A\n\nB, the induced map from GL2(A) to GL2(B) will also be\n\nâ†’\n\ndenoted Ï†.