Modular Lifting Conjecture, but this time there is no analogue of the Langlands-Tunnell Theorem to help. Wiles uses the Hilbert Irreducibility Theorem, applied to a parameter space of elliptic curves, to produce another semistable elliptic curve Eâ€² over Q satisfying\n\n(i) Â¯Ï�Eâ€²,5 is isomorphic to Â¯Ï�E,5, and (ii) Â¯Ï�Eâ€²,3 is irreducible.\n\n(In fact there will be inï¬�nitely many such Eâ€²; see Appendix B.2.) Now by Proposi- âˆ� n=1 ane2Ï€inz be a corresponding eigenform. tion 2.3, Eâ€² is modular. Let f (z) = Then for all but ï¬�nitely many primes q, P\n\n#(Eâ€²(Fq)) trace(Â¯Ï�E,5(Frobq))\n\naq = q + 1\n\ntrace(Â¯Ï�Eâ€²,5(Frobq)) #(E(Fq)) q + 1\n\nâˆ’\n\nâ‰¡ â‰¡\n\n(mod 5)\n\nâ‰¡\n\nâˆ’\n\nby (7). Thus the form f satisï¬�es hypothesis (b) of the Semistable Modular Lifting Conjecture, and we conclude that E is modular.\n\nTaken together, Propositions 2.3 and 2.4 show that the Semistable Modular Lifting Conjecture for p = 3 and 5 implies the Semistable Taniyama-Shimura Con- jecture.\n\n3. Galois representations\n\nThe next step is to translate the Semistable Modular Lifting Conjecture into a conjecture (Conjecture 3.2) about the modularity of liftings of Galois repre- sentations. Throughout this paper, if A is a topological ring, a representation GL2(A) will mean a continuous homomorphism and [Ï�] will denote the Ï� : GQ isomorphism class of Ï�. If p is a prime, let\n\nâ†’\n\nZÃ— p\n\nÎµp : GQ\n\nâ†’\n\nbe the character giving the action of GQ on p-power roots of unity.\n\n10\n\nK. RUBIN AND A. SILVERBERG\n\n3.1. The p-adic representation attached to an elliptic curve. Suppose E is an elliptic curve over Q and p is a prime number. For every positive integer n, write E[pn] for the subgroup in E( Â¯Q) of points of order dividing pn and Tp(E) for the inverse limit of the E[pn] with respect to multiplication by p. For every n, E[pn] âˆ¼= (Z/pnZ)2, and so Tp(E) âˆ¼= Z2 p. The action of GQ induces a representation\n\nGL2(Zp)\n\nÏ�E,p : GQ\n\nâ†’\n\nsuch that
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: Modular Lifting Conjecture, but this time there is no analogue of the Langlands-Tunnell Theorem to help. Wiles uses the Hilbert Irreducibility Theorem, applied to a parameter space of elliptic curves, to produce another semistable elliptic curve Eâ€² over Q satisfying\n\n(i) Â¯Ï�Eâ€²,5 is isomorphic to Â¯Ï�E,5, and (ii) Â¯Ï�Eâ€²,3 is irreducible.\n\n(In fact there will be inï¬�nitely many such Eâ€²; see Appendix B.2.) Now by Proposi- âˆ� n=1 ane2Ï€inz be a corresponding eigenform. tion 2.3, Eâ€² is modular. Let f (z) = Then for all but ï¬�nitely many primes q, P\n\n#(Eâ€²(Fq)) trace(Â¯Ï�E,5(Frobq))\n\naq = q + 1\n\ntrace(Â¯Ï�Eâ€²,5(Frobq)) #(E(Fq)) q + 1\n\nâˆ’\n\nâ‰¡ â‰¡\n\n(mod 5)\n\nâ‰¡\n\nâˆ’\n\nby (7). Thus the form f satisï¬�es hypothesis (b) of the Semistable Modular Lifting Conjecture, and we conclude that E is modular.\n\nTaken together, Propositions 2.3 and 2.4 show that the Semistable Modular Lifting Conjecture for p = 3 and 5 implies the Semistable Taniyama-Shimura Con- jecture.\n\n3. Galois representations\n\nThe next step is to translate the Semistable Modular Lifting Conjecture into a conjecture (Conjecture 3.2) about the modularity of liftings of Galois repre- sentations. Throughout this paper, if A is a topological ring, a representation GL2(A) will mean a continuous homomorphism and [Ï�] will denote the Ï� : GQ isomorphism class of Ï�. If p is a prime, let\n\nâ†’\n\nZÃ— p\n\nÎµp : GQ\n\nâ†’\n\nbe the character giving the action of GQ on p-power roots of unity.\n\n10\n\nK. RUBIN AND A. SILVERBERG\n\n3.1. The p-adic representation attached to an elliptic curve. Suppose E is an elliptic curve over Q and p is a prime number. For every positive integer n, write E[pn] for the subgroup in E( Â¯Q) of points of order dividing pn and Tp(E) for the inverse limit of the E[pn] with respect to multiplication by p. For every n, E[pn] âˆ¼= (Z/pnZ)2, and so Tp(E) âˆ¼= Z2 p. The action of GQ induces a representation\n\nGL2(Zp)\n\nÏ�E,p : GQ\n\nâ†’\n\nsuch that