and Fermatâ€™s Last Theorem. Wilesâ€™ proof of Theorem 5.3 depends on the not-yet-complete construction of an appropriate Euler system (as in 5.4), while his proof of Theorem 5.4 (though not yet fully checked) does not. For Theorem 5.4, Wiles bounds the Selmer group of 5.2 without constructing a new Euler system, by using results from the Iwasawa theory of imaginary quadratic ï¬�elds. (These results in turn rely on Kolyvaginâ€™s method and the Euler system of elliptic units; see [31].)\n\nÂ§\n\nÂ§\n\nSince for ease of exposition we deï¬�ned modularity of representations in terms of Î“0(N ) instead of Î“1(N ), the theorems stated below are weaker than those an- nounced by Wiles, but have the same applications to elliptic curves. (Note that by our deï¬�nition of type-\n\n, if Â¯Ï� is type-\n\n, then det(Â¯Ï�) = Â¯Îµp.)\n\nD\n\nD\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n19\n\nIf Â¯Ï� is a representation of GQ on a vector space V , Sym2(Â¯Ï�) denotes the repre-\n\nsentation on the symmetric square of V induced by Â¯Ï�.\n\nTheorem 5.3 (Wiles). Suppose p, k, the following additional conditions :\n\n, Â¯Ï�, and\n\nare as in\n\n4.2 and Â¯Ï� satisï¬�es\n\nD\n\nO\n\nÂ§\n\n(i) Sym2(Â¯Ï�) is absolutely irreducible, (ii) if Â¯Ï� is ramiï¬�ed at q and q (iii) if p is 3 or 5, then for some prime q, p divides #(Â¯Ï�(Iq)).\n\n= p, then the restriction of Â¯Ï� to Dq is reducible,\n\n6\n\nT is an isomorphism.\n\nThen Ï• : R\n\nâ†’\n\nSince Theorem 5.3 does not yield the full Mazur Conjecture (Conjecture 4.2) for 2 to see which elliptic curves Â§\n\np = 3 and 5, we need to reexamine the arguments of E can be proved modular using Theorem 5.3 applied to Â¯Ï�E,3 and Â¯Ï�E,5.\n\nHypothesis (i) of Theorem 5.3 will be satisï¬�ed if the image of Â¯Ï�E,p is suï¬ƒciently large in GL2(Fp) (for example, if Â¯Ï�E,p is surjective). For p = 3 and p = 5, if Â¯Ï�E,p satisï¬�es hypothesis (iii) and is irreducible, then it satisï¬�es hypothesis (i).\n\nIf E is semistable, p is an odd prime, and
Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place.
Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: and Fermatâ€™s Last Theorem. Wilesâ€™ proof of Theorem 5.3 depends on the not-yet-complete construction of an appropriate Euler system (as in 5.4), while his proof of Theorem 5.4 (though not yet fully checked) does not. For Theorem 5.4, Wiles bounds the Selmer group of 5.2 without constructing a new Euler system, by using results from the Iwasawa theory of imaginary quadratic ï¬�elds. (These results in turn rely on Kolyvaginâ€™s method and the Euler system of elliptic units; see [31].)\n\nÂ§\n\nÂ§\n\nSince for ease of exposition we deï¬�ned modularity of representations in terms of Î“0(N ) instead of Î“1(N ), the theorems stated below are weaker than those an- nounced by Wiles, but have the same applications to elliptic curves. (Note that by our deï¬�nition of type-\n\n, if Â¯Ï� is type-\n\n, then det(Â¯Ï�) = Â¯Îµp.)\n\nD\n\nD\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n19\n\nIf Â¯Ï� is a representation of GQ on a vector space V , Sym2(Â¯Ï�) denotes the repre-\n\nsentation on the symmetric square of V induced by Â¯Ï�.\n\nTheorem 5.3 (Wiles). Suppose p, k, the following additional conditions :\n\n, Â¯Ï�, and\n\nare as in\n\n4.2 and Â¯Ï� satisï¬�es\n\nD\n\nO\n\nÂ§\n\n(i) Sym2(Â¯Ï�) is absolutely irreducible, (ii) if Â¯Ï� is ramiï¬�ed at q and q (iii) if p is 3 or 5, then for some prime q, p divides #(Â¯Ï�(Iq)).\n\n= p, then the restriction of Â¯Ï� to Dq is reducible,\n\n6\n\nT is an isomorphism.\n\nThen Ï• : R\n\nâ†’\n\nSince Theorem 5.3 does not yield the full Mazur Conjecture (Conjecture 4.2) for 2 to see which elliptic curves Â§\n\np = 3 and 5, we need to reexamine the arguments of E can be proved modular using Theorem 5.3 applied to Â¯Ï�E,3 and Â¯Ï�E,5.\n\nHypothesis (i) of Theorem 5.3 will be satisï¬�ed if the image of Â¯Ï�E,p is suï¬ƒciently large in GL2(Fp) (for example, if Â¯Ï�E,p is surjective). For p = 3 and p = 5, if Â¯Ï�E,p satisï¬�es hypothesis (iii) and is irreducible, then it satisï¬�es hypothesis (i).\n\nIf E is semistable, p is an odd prime, and