of the complex numbers, there is a natural commutative group law on the set of F -solutions of E, with the point at inï¬�nity as the identity element. Denote this group E(F ). If p is a prime, write E[p] for the subgroup of points in E( Â¯Q) of order dividing p. Then E[p] âˆ¼= F2 p. The action of GQ on E[p] gives a continuous representation\n\nGL2(Fp)\n\nÂ¯Ï�E,p : GQ\n\nâ†’\n\n(deï¬�ned up to isomorphism) such that\n\n(6)\n\ndet(Â¯Ï�E,p) = Â¯Îµp\n\nand for all but ï¬�nitely many primes q,\n\n#(E(Fq))\n\n(7)\n\ntrace(Â¯Ï�E,p(Frobq))\n\nq + 1\n\n(mod p).\n\nâ‰¡ (See Appendix A for the deï¬�nition of the Frobenius elements Frobq âˆˆ to each prime number q.)\n\nâˆ’\n\nGQ attached\n\nâˆ� n=1 ane2Ï€inz is an eigenform, let\n\nOf denote the ring of integers of the number ï¬�eld Q(a2, a3, . . . ). (Recall that our eigenforms are normalized so that a1 = 1.)\n\nIf f (z) =\n\nP\n\nThe following conjecture is in the spirit of a conjecture of Mazur (see Conjectures\n\n3.2 and 4.2).\n\nConjecture 2.1 (Semistable Modular Lifting Conjecture). Suppose p is an odd prime and E is a semistable elliptic curve over Q satisfying\n\n(a) Â¯Ï�E,p is irreducible, (b) there are an eigenform f (z) =\n\nâˆ� n=1 ane2Ï€inz and a prime ideal Î» of\n\nOf\n\nsuch that p\n\nÎ» and for all but ï¬�nitely many primes q,\n\nâˆˆ\n\nP\n\n#(E(Fq))\n\naq â‰¡\n\nq + 1\n\n(mod Î»).\n\nâˆ’\n\nThen E is modular.\n\nThe Semistable Modular Lifting Conjecture is a priori weaker than the Semi- stable Taniyama-Shimura Conjecture because of the extra hypotheses (a) and (b). The more serious condition is (b); there is no known way to produce such a form in general. But when p = 3, the existence of such a form follows from the theorem below of Tunnell [41] and Langlands [20]. Wiles then gets around condition (a) by a clever argument (described below) which, when Â¯Ï�E,3 is not irreducible, allows him to use p = 5 instead.\n\n8\n\nK. RUBIN AND A. SILVERBERG\n\n2.2. Langlands-Tunnell Theorem. In order to state 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. ->: of the complex numbers, there is a natural commutative group law on the set of F -solutions of E, with the point at inï¬�nity as the identity element. Denote this group E(F ). If p is a prime, write E[p] for the subgroup of points in E( Â¯Q) of order dividing p. Then E[p] âˆ¼= F2 p. The action of GQ on E[p] gives a continuous representation\n\nGL2(Fp)\n\nÂ¯Ï�E,p : GQ\n\nâ†’\n\n(deï¬�ned up to isomorphism) such that\n\n(6)\n\ndet(Â¯Ï�E,p) = Â¯Îµp\n\nand for all but ï¬�nitely many primes q,\n\n#(E(Fq))\n\n(7)\n\ntrace(Â¯Ï�E,p(Frobq))\n\nq + 1\n\n(mod p).\n\nâ‰¡ (See Appendix A for the deï¬�nition of the Frobenius elements Frobq âˆˆ to each prime number q.)\n\nâˆ’\n\nGQ attached\n\nâˆ� n=1 ane2Ï€inz is an eigenform, let\n\nOf denote the ring of integers of the number ï¬�eld Q(a2, a3, . . . ). (Recall that our eigenforms are normalized so that a1 = 1.)\n\nIf f (z) =\n\nP\n\nThe following conjecture is in the spirit of a conjecture of Mazur (see Conjectures\n\n3.2 and 4.2).\n\nConjecture 2.1 (Semistable Modular Lifting Conjecture). Suppose p is an odd prime and E is a semistable elliptic curve over Q satisfying\n\n(a) Â¯Ï�E,p is irreducible, (b) there are an eigenform f (z) =\n\nâˆ� n=1 ane2Ï€inz and a prime ideal Î» of\n\nOf\n\nsuch that p\n\nÎ» and for all but ï¬�nitely many primes q,\n\nâˆˆ\n\nP\n\n#(E(Fq))\n\naq â‰¡\n\nq + 1\n\n(mod Î»).\n\nâˆ’\n\nThen E is modular.\n\nThe Semistable Modular Lifting Conjecture is a priori weaker than the Semi- stable Taniyama-Shimura Conjecture because of the extra hypotheses (a) and (b). The more serious condition is (b); there is no known way to produce such a form in general. But when p = 3, the existence of such a form follows from the theorem below of Tunnell [41] and Langlands [20]. Wiles then gets around condition (a) by a clever argument (described below) which, when Â¯Ï�E,3 is not irreducible, allows him to use p = 5 instead.\n\n8\n\nK. RUBIN AND A. SILVERBERG\n\n2.2. Langlands-Tunnell Theorem. In order to state the