on the order of the cotangent space at a prime of R. In 5.2 we Â§ see that the corresponding tangent space is a Selmer group, and in 5.3 we outline a general procedure due to Kolyvagin for bounding sizes of Selmer groups. The input for Kolyvaginâ€™s method is known as an Euler system. The most diï¬ƒcult 5.4), and the part described as â€œnot yet completeâ€� in his part of Wilesâ€™ work ( Â§ December announcement, is his construction of a suitable Euler system. In 5.5 we state the results announced by Wiles (Theorems 5.3 and 5.4 and Corollary 5.5) and explain why Theorem 5.3 suï¬ƒces for proving the Semistable Taniyama-Shimura Conjecture. As an application of Corollary 5.5 we write down an inï¬�nite family of modular elliptic curves. , Â¯Ï�, 5 ï¬�x p, k,\n\nÂ§\n\nÂ§\n\nâˆ� n=1 ane2Ï€inz, and Î» as in\n\n4.2.\n\nFor O By property (T4) there is a homomorphism\n\n, f (z) =\n\nÂ§\n\nÂ§\n\nD\n\nP\n\nÏ€ : T\n\nâ†’ O . By property (T2) and (12), Ï€ satisï¬�es\n\nsuch that Ï€ Ï€(Tq) = aq for all but ï¬�nitely many q.\n\nÏ�T is isomorphic to Ï�f,Î» âŠ— O\n\n\n\n5.1. Key reduction. Wiles uses the following generalization of a theorem of Mazur, which says that T is Gorenstein.\n\nTheorem 5.1. There is a (noncanonical ) T-module isomorphism\n\n) âˆ¼ â†’\n\nHomO(T,\n\nT.\n\nO\n\nLet Î· denote the ideal of\n\ngenerated by the image under the composition\n\nO HomO(T,\n\n) âˆ¼ â†’\n\nT Ï€\n\nO\n\nâ†’ O\n\nHomO(T,\n\nof the element Ï€ âˆˆ choice of isomorphism in Theorem 5.1.\n\n). The ideal Î· is well deï¬�ned independent of the\n\nO\n\nThe map Ï€ determines distinguished prime ideals of T and R,\n\nÏ•) = Ï•âˆ’1(pT).\n\npT = ker(Ï€),\n\npR = ker(Ï€\n\n\n\nTheorem 5.2 (Wiles). If\n\n#(pR/p2\n\nR)\n\n#(\n\n/Î·) <\n\n, âˆ�\n\nâ‰¤\n\nO\n\nT is an isomorphism.\n\nthen Ï• : R\n\nâ†’\n\nThe proof is entirely commutative algebra. The surjectivity of Ï• shows that /Î·). Thus if\n\n#(pR/p2 #(pR/p2\n\n#(pT/p2 #(\n\nT), and Wiles proves that #(pT/p2\n\nR) R)\n\nT)\n\n#(\n\nâ‰¥
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: on the order of the cotangent space at a prime of R. In 5.2 we Â§ see that the corresponding tangent space is a Selmer group, and in 5.3 we outline a general procedure due to Kolyvagin for bounding sizes of Selmer groups. The input for Kolyvaginâ€™s method is known as an Euler system. The most diï¬ƒcult 5.4), and the part described as â€œnot yet completeâ€� in his part of Wilesâ€™ work ( Â§ December announcement, is his construction of a suitable Euler system. In 5.5 we state the results announced by Wiles (Theorems 5.3 and 5.4 and Corollary 5.5) and explain why Theorem 5.3 suï¬ƒces for proving the Semistable Taniyama-Shimura Conjecture. As an application of Corollary 5.5 we write down an inï¬�nite family of modular elliptic curves. , Â¯Ï�, 5 ï¬�x p, k,\n\nÂ§\n\nÂ§\n\nâˆ� n=1 ane2Ï€inz, and Î» as in\n\n4.2.\n\nFor O By property (T4) there is a homomorphism\n\n, f (z) =\n\nÂ§\n\nÂ§\n\nD\n\nP\n\nÏ€ : T\n\nâ†’ O . By property (T2) and (12), Ï€ satisï¬�es\n\nsuch that Ï€ Ï€(Tq) = aq for all but ï¬�nitely many q.\n\nÏ�T is isomorphic to Ï�f,Î» âŠ— O\n\n\n\n5.1. Key reduction. Wiles uses the following generalization of a theorem of Mazur, which says that T is Gorenstein.\n\nTheorem 5.1. There is a (noncanonical ) T-module isomorphism\n\n) âˆ¼ â†’\n\nHomO(T,\n\nT.\n\nO\n\nLet Î· denote the ideal of\n\ngenerated by the image under the composition\n\nO HomO(T,\n\n) âˆ¼ â†’\n\nT Ï€\n\nO\n\nâ†’ O\n\nHomO(T,\n\nof the element Ï€ âˆˆ choice of isomorphism in Theorem 5.1.\n\n). The ideal Î· is well deï¬�ned independent of the\n\nO\n\nThe map Ï€ determines distinguished prime ideals of T and R,\n\nÏ•) = Ï•âˆ’1(pT).\n\npT = ker(Ï€),\n\npR = ker(Ï€\n\n\n\nTheorem 5.2 (Wiles). If\n\n#(pR/p2\n\nR)\n\n#(\n\n/Î·) <\n\n, âˆ�\n\nâ‰¤\n\nO\n\nT is an isomorphism.\n\nthen Ï• : R\n\nâ†’\n\nThe proof is entirely commutative algebra. The surjectivity of Ï• shows that /Î·). Thus if\n\n#(pR/p2 #(pR/p2\n\n#(pT/p2 #(\n\nT), and Wiles proves that #(pT/p2\n\nR) R)\n\nT)\n\n#(\n\nâ‰¥