Selmer group is an important tool in studying the arithmetic of E because it contains (via (14)) E(Q)/mE(Q).\n\n5, let m denote the maximal ideal /mn) can be\n\nRetaining the notation from the beginning of\n\nÂ§\n\nand ï¬�x a positive integer n. The tangent space HomO(pR/p2 R,\n\nof identiï¬�ed with a Selmer group as follows. Let Vn be the matrix algebra M2(\n\nO\n\nO\n\n/mn), with GQ acting via the adjoint repre-\n\nO\n\nsentation Ïƒ(B) = Ï�f,Î»(Ïƒ)BÏ�f,Î»(Ïƒ)âˆ’1. There is a natural injection\n\ns : HomO(pR/p2 R,\n\n/mn) Ö’\n\nH 1(Q, Vn)\n\nO\n\nâ†’\n\nwhich is described in Appendix D (see also\n\n1.6 of [21]). Wiles deï¬�nes a collection . Let SD(Vn) denote the associated Selmer\n\nÂ§\n\nH 1(Qq, Vn) }\n\n=\n\nJq âŠ†\n\ndepending on\n\nJ group. Wiles proves that s induces an isomorphism\n\n{\n\nD\n\n/mn) âˆ¼ â†’\n\nHomO(pR/p2 R,\n\nSD(Vn).\n\nO\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n17\n\n5.3. Euler systems. We have now reduced the proof of Mazurâ€™s conjecture to bounding the size of the Selmer groups SD(Vn). About ï¬�ve years ago Kolyvagin [19], building on ideas of his own and of Thaine [40], introduced a revolutionary new method for bounding the size of a Selmer group. This new machinery, which is crucial for Wilesâ€™ proof, is what we now describe.\n\nH 1(Qq,M ) is } 5.2. Let Ë†M = a system of subgroups with associated Selmer group S(M ) as in Hom(M, Âµm), where Âµm is the group of m-th roots of unity. For every prime q, the cup product gives a nondegenerate Tate pairing\n\nSuppose M is a GQ-module of odd exponent m and\n\n=\n\nJq âŠ† Â§\n\nJ\n\n{\n\nH 2(Qq, Âµm) âˆ¼ â†’ H 1(Q, Ë†M ), then\n\nH 1(Qq, Ë†M )\n\niq : H 1(Qq, M )\n\nZ/mZ\n\n,\n\nh\n\nÃ—\n\nâ†’\n\nH 1(Q, M ) and d\n\n(see Chapters VI and VII of [3]). If c\n\nâˆˆ\n\nâˆˆ\n\n(15)\n\nresq(c), resq(d) h\n\niq = 0.\n\nq X\n\nH 1(Q, Ë†M ) be the Selmer\n\nis a ï¬�nite set of primes. Let Sâˆ—\n\nSuppose that\n\nL âŠ† H 1(Qq, Ë†M ) }\n\nL group given by the local conditions\n\nâˆ— =\n\nJ âˆ— q
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: Selmer group is an important tool in studying the arithmetic of E because it contains (via (14)) E(Q)/mE(Q).\n\n5, let m denote the maximal ideal /mn) can be\n\nRetaining the notation from the beginning of\n\nÂ§\n\nand ï¬�x a positive integer n. The tangent space HomO(pR/p2 R,\n\nof identiï¬�ed with a Selmer group as follows. Let Vn be the matrix algebra M2(\n\nO\n\nO\n\n/mn), with GQ acting via the adjoint repre-\n\nO\n\nsentation Ïƒ(B) = Ï�f,Î»(Ïƒ)BÏ�f,Î»(Ïƒ)âˆ’1. There is a natural injection\n\ns : HomO(pR/p2 R,\n\n/mn) Ö’\n\nH 1(Q, Vn)\n\nO\n\nâ†’\n\nwhich is described in Appendix D (see also\n\n1.6 of [21]). Wiles deï¬�nes a collection . Let SD(Vn) denote the associated Selmer\n\nÂ§\n\nH 1(Qq, Vn) }\n\n=\n\nJq âŠ†\n\ndepending on\n\nJ group. Wiles proves that s induces an isomorphism\n\n{\n\nD\n\n/mn) âˆ¼ â†’\n\nHomO(pR/p2 R,\n\nSD(Vn).\n\nO\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n17\n\n5.3. Euler systems. We have now reduced the proof of Mazurâ€™s conjecture to bounding the size of the Selmer groups SD(Vn). About ï¬�ve years ago Kolyvagin [19], building on ideas of his own and of Thaine [40], introduced a revolutionary new method for bounding the size of a Selmer group. This new machinery, which is crucial for Wilesâ€™ proof, is what we now describe.\n\nH 1(Qq,M ) is } 5.2. Let Ë†M = a system of subgroups with associated Selmer group S(M ) as in Hom(M, Âµm), where Âµm is the group of m-th roots of unity. For every prime q, the cup product gives a nondegenerate Tate pairing\n\nSuppose M is a GQ-module of odd exponent m and\n\n=\n\nJq âŠ† Â§\n\nJ\n\n{\n\nH 2(Qq, Âµm) âˆ¼ â†’ H 1(Q, Ë†M ), then\n\nH 1(Qq, Ë†M )\n\niq : H 1(Qq, M )\n\nZ/mZ\n\n,\n\nh\n\nÃ—\n\nâ†’\n\nH 1(Q, M ) and d\n\n(see Chapters VI and VII of [3]). If c\n\nâˆˆ\n\nâˆˆ\n\n(15)\n\nresq(c), resq(d) h\n\niq = 0.\n\nq X\n\nH 1(Q, Ë†M ) be the Selmer\n\nis a ï¬�nite set of primes. Let Sâˆ—\n\nSuppose that\n\nL âŠ† H 1(Qq, Ë†M ) }\n\nL group given by the local conditions\n\nâˆ— =\n\nJ âˆ— q