3).\n\nâ‰¡\n\nExplicitly, Ïˆ can be deï¬�ned on generators of GL2(F3) by\n\nâˆš\n\n1 1 1 0\n\n1 1 1 0\n\n1 1\n\n1 1\n\n2 1 1 0\n\n.\n\nÏˆ\n\n=\n\nand Ïˆ\n\n=\n\nâˆ’ âˆ’\n\nâˆ’ âˆ’\n\nâˆ’\n\nâˆ’\n\n(cid:19)\n\n(cid:18)(cid:18)\n\n(cid:19)(cid:19)\n\n(cid:18)\n\n(cid:18)(cid:18) Â¯Ï�E,3. If Ï„ is complex conjugation, then it follows from (6) and (10) that 1. The image of Ïˆ in PGL2(C) is a subgroup of PGL2(F3) âˆ¼= S4.\n\n(cid:19)\n\n(cid:19)(cid:19)\n\n(cid:18)\n\nLet Ï� = Ïˆ â—¦ det(Ï�(Ï„ )) = Using that Â¯Ï�E,3 is irreducible, one can show that Ï� is irreducible.\n\nâˆ’\n\nâˆ� n=1 bne2Ï€inz be a weight-one cusp form for some Î“1(N ) obtained by applying the Langlands-Tunnell\n\nLet p be a prime of Â¯Q containing 1 + âˆš\n\n2. Let g(z) =\n\nâˆ’\n\nP\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n9\n\nTheorem (Theorem 2.2) to Ï�. It follows from (6) and (10) that N is divisible by 3. The function\n\n0 if d 1 if d 1 if d\n\n0 (mod 3), 1 (mod 3), 2 (mod 3)\n\nâˆ�\n\nâ‰¡ â‰¡ â‰¡\n\nÏ‡(d)e2Ï€inz where Ï‡(d) =\n\nE(z) = 1 + 6\n\n\uf8f1 \uf8f2\n\nn=1 X\n\nXd|n\n\nâˆ’\n\nâˆ� n=1 cne2Ï€inz is a weight-one modular form for Î“1(3). The product g(z)E(z) = It is now is a weight-two cusp form for Î“0(N ) with cn â‰¡ bn possible to ï¬�nd an eigenform f (z) = (mod p) for every n (see 6.10 and 6.11 of [4]). By (7), (8), and (9), f satisï¬�es (b) of the Semistable Modular Lifting Conjecture with p = 3 and with Î» = p\n\n\uf8f3\n\nbn (mod p) for all n. P n=1 ane2Ï€inz on Î“0(N ) such that an â‰¡ âˆ© Of .\n\nâˆ�\n\nP\n\nProposition 2.4 (Wiles). Suppose the Semistable Modular Lifting Conjecture is true for p = 3 and 5, E is a semistable elliptic curve over Q, and Â¯Ï�E,3 is reducible. Then E is modular.\n\nProof. The elliptic curves over Q for which both Â¯Ï�E,3 and Â¯Ï�E,5 are reducible are all known to be modular (see Appendix B.1). Thus we can suppose Â¯Ï�E,5 is irreducible. It suï¬ƒces to produce an eigenform as in (b) of the Semistable Modular Lifting Conjecture, but this time
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: 3).\n\nâ‰¡\n\nExplicitly, Ïˆ can be deï¬�ned on generators of GL2(F3) by\n\nâˆš\n\n1 1 1 0\n\n1 1 1 0\n\n1 1\n\n1 1\n\n2 1 1 0\n\n.\n\nÏˆ\n\n=\n\nand Ïˆ\n\n=\n\nâˆ’ âˆ’\n\nâˆ’ âˆ’\n\nâˆ’\n\nâˆ’\n\n(cid:19)\n\n(cid:18)(cid:18)\n\n(cid:19)(cid:19)\n\n(cid:18)\n\n(cid:18)(cid:18) Â¯Ï�E,3. If Ï„ is complex conjugation, then it follows from (6) and (10) that 1. The image of Ïˆ in PGL2(C) is a subgroup of PGL2(F3) âˆ¼= S4.\n\n(cid:19)\n\n(cid:19)(cid:19)\n\n(cid:18)\n\nLet Ï� = Ïˆ â—¦ det(Ï�(Ï„ )) = Using that Â¯Ï�E,3 is irreducible, one can show that Ï� is irreducible.\n\nâˆ’\n\nâˆ� n=1 bne2Ï€inz be a weight-one cusp form for some Î“1(N ) obtained by applying the Langlands-Tunnell\n\nLet p be a prime of Â¯Q containing 1 + âˆš\n\n2. Let g(z) =\n\nâˆ’\n\nP\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n9\n\nTheorem (Theorem 2.2) to Ï�. It follows from (6) and (10) that N is divisible by 3. The function\n\n0 if d 1 if d 1 if d\n\n0 (mod 3), 1 (mod 3), 2 (mod 3)\n\nâˆ�\n\nâ‰¡ â‰¡ â‰¡\n\nÏ‡(d)e2Ï€inz where Ï‡(d) =\n\nE(z) = 1 + 6\n\n\uf8f1 \uf8f2\n\nn=1 X\n\nXd|n\n\nâˆ’\n\nâˆ� n=1 cne2Ï€inz is a weight-one modular form for Î“1(3). The product g(z)E(z) = It is now is a weight-two cusp form for Î“0(N ) with cn â‰¡ bn possible to ï¬�nd an eigenform f (z) = (mod p) for every n (see 6.10 and 6.11 of [4]). By (7), (8), and (9), f satisï¬�es (b) of the Semistable Modular Lifting Conjecture with p = 3 and with Î» = p\n\n\uf8f3\n\nbn (mod p) for all n. P n=1 ane2Ï€inz on Î“0(N ) such that an â‰¡ âˆ© Of .\n\nâˆ�\n\nP\n\nProposition 2.4 (Wiles). Suppose the Semistable Modular Lifting Conjecture is true for p = 3 and 5, E is a semistable elliptic curve over Q, and Â¯Ï�E,3 is reducible. Then E is modular.\n\nProof. The elliptic curves over Q for which both Â¯Ï�E,3 and Â¯Ï�E,5 are reducible are all known to be modular (see Appendix B.1). Thus we can suppose Â¯Ï�E,5 is irreducible. It suï¬ƒces to produce an eigenform as in (b) of the Semistable Modular Lifting Conjecture, but this time