Langlands-Tunnell Theorem. In order to state the Langlands-Tunnell Theorem, we need weight-one modular forms for a subgroup of Î“0(N ). Let\n\na b c d\n\nSL2(Z) : c\n\n0 (mod N ), a\n\nd\n\n1 (mod N )\n\n.\n\nÎ“1(N ) =\n\nâˆˆ\n\nâ‰¡\n\nâ‰¡\n\nâ‰¡\n\n(cid:1)\n\n(cid:9)\n\n(cid:8)(cid:0)\n\nReplacing Î“0(N ) by Î“1(N ) in 1.6, one can deï¬�ne the notion of cusp forms on Â§ Î“1(N ). See Chapter 3 of [35] for the deï¬�nitions of the Hecke operators on the space of weight-one cusp forms for Î“1(N ).\n\nTheorem 2.2 (Langlands-Tunnell). Suppose Ï� : GQ GL2(C) is a continuous irreducible representation whose image in PGL2(C) is a subgroup of S4 (the sym- metric group on four elements ), Ï„ is complex conjugation, and det(Ï�(Ï„ )) = 1. âˆ� n=1 bne2Ï€inz for some Î“1(N ), which is an Then there is a weight-one cusp form eigenfunction for all the corresponding Hecke operators, such that for all but ï¬�nitely many primes q,\n\nâ†’\n\nâˆ’\n\nP\n\n(8)\n\nbq = trace(Ï�(Frobq)).\n\nThe theorem as stated by Langlands [20] and by Tunnell [41] produces an auto- morphic representation rather than a cusp form. Using the fact that det(Ï�(Ï„ )) = 1, standard techniques (see for example [12]) show that this automorphic repre-\n\nâˆ’ sentation corresponds to a weight-one cusp form as in Theorem 2.2.\n\n2.3. Semistable Modular Lifting\n\nSemistable Taniyama-Shimura.\n\nâ‡’\n\nProposition 2.3. Suppose the Semistable Modular Lifting Conjecture is true for p = 3, E is a semistable elliptic curve, and Â¯Ï�E,3 is irreducible. Then E is modular.\n\nProof. It suï¬ƒces to show that hypothesis (b) of the Semistable Modular Lifting Conjecture is satisï¬�ed with the given curve E, for p = 3. There is a faithful representation\n\nGL2(Z[âˆš\n\nGL2(C)\n\nÏˆ : GL2(F3) Ö’\n\n2])\n\nâˆ’\n\nâŠ‚\n\nâ†’\n\nGL2(F3),\n\nsuch that for every g\n\nâˆˆ trace(Ïˆ(g))\n\n(mod(1 + âˆš\n\n(9)\n\ntrace(g)\n\n2))\n\nâ‰¡\n\nâˆ’\n\nand\n\n(10)\n\ndet(Ïˆ(g))\n\ndet(g)\n\n(mod 3).\n\nâ‰¡\n\nExplicitly, Ïˆ can be deï¬�ned on
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: Langlands-Tunnell Theorem. In order to state the Langlands-Tunnell Theorem, we need weight-one modular forms for a subgroup of Î“0(N ). Let\n\na b c d\n\nSL2(Z) : c\n\n0 (mod N ), a\n\nd\n\n1 (mod N )\n\n.\n\nÎ“1(N ) =\n\nâˆˆ\n\nâ‰¡\n\nâ‰¡\n\nâ‰¡\n\n(cid:1)\n\n(cid:9)\n\n(cid:8)(cid:0)\n\nReplacing Î“0(N ) by Î“1(N ) in 1.6, one can deï¬�ne the notion of cusp forms on Â§ Î“1(N ). See Chapter 3 of [35] for the deï¬�nitions of the Hecke operators on the space of weight-one cusp forms for Î“1(N ).\n\nTheorem 2.2 (Langlands-Tunnell). Suppose Ï� : GQ GL2(C) is a continuous irreducible representation whose image in PGL2(C) is a subgroup of S4 (the sym- metric group on four elements ), Ï„ is complex conjugation, and det(Ï�(Ï„ )) = 1. âˆ� n=1 bne2Ï€inz for some Î“1(N ), which is an Then there is a weight-one cusp form eigenfunction for all the corresponding Hecke operators, such that for all but ï¬�nitely many primes q,\n\nâ†’\n\nâˆ’\n\nP\n\n(8)\n\nbq = trace(Ï�(Frobq)).\n\nThe theorem as stated by Langlands [20] and by Tunnell [41] produces an auto- morphic representation rather than a cusp form. Using the fact that det(Ï�(Ï„ )) = 1, standard techniques (see for example [12]) show that this automorphic repre-\n\nâˆ’ sentation corresponds to a weight-one cusp form as in Theorem 2.2.\n\n2.3. Semistable Modular Lifting\n\nSemistable Taniyama-Shimura.\n\nâ‡’\n\nProposition 2.3. Suppose the Semistable Modular Lifting Conjecture is true for p = 3, E is a semistable elliptic curve, and Â¯Ï�E,3 is irreducible. Then E is modular.\n\nProof. It suï¬ƒces to show that hypothesis (b) of the Semistable Modular Lifting Conjecture is satisï¬�ed with the given curve E, for p = 3. There is a faithful representation\n\nGL2(Z[âˆš\n\nGL2(C)\n\nÏˆ : GL2(F3) Ö’\n\n2])\n\nâˆ’\n\nâŠ‚\n\nâ†’\n\nGL2(F3),\n\nsuch that for every g\n\nâˆˆ trace(Ïˆ(g))\n\n(mod(1 + âˆš\n\n(9)\n\ntrace(g)\n\n2))\n\nâ‰¡\n\nâˆ’\n\nand\n\n(10)\n\ndet(Ïˆ(g))\n\ndet(g)\n\n(mod 3).\n\nâ‰¡\n\nExplicitly, Ïˆ can be deï¬�ned on