proves that #(pT/p2\n\nR) R)\n\nT)\n\n#(\n\nâ‰¥ â‰¤\n\nâ‰¥\n\nO\n\n/Î·), then\n\nO\n\n#(pR/p2\n\nR) = #(pT/p2\n\n(13)\n\nT) = #(\n\n/Î·).\n\nO\n\nThe ï¬�rst equality in (13) shows that Ï• induces an isomorphism of tangent spaces. Wiles uses the second equality in (13) and Theorem 5.1 to deduce that T is a local\n\n16\n\nK. RUBIN AND A. SILVERBERG\n\ncomplete intersection over that\n\n(that is, there are f1, . . . , fr âˆˆ O\n\n[[x1, . . . , xr]] such\n\nO\n\nT âˆ¼=\n\n[[x1, . . . , xr]]/(f1, . . . , fr)\n\nO\n\nas morphism.\n\nalgebras). Wiles then combines these two results to prove that Ï• is an iso-\n\nO\n\n5.2. Selmer groups. In general, if M is a torsion GQ-module, a Selmer group attached to M is a subgroup of the Galois cohomology group H 1(GQ, M ) deter- mined by certain â€œlocal conditionsâ€� in the following way. If q is a prime with decomposition group Dq âŠ‚\n\nGQ, then there is a restriction map\n\nresq : H 1(GQ, M )\n\nH 1(Dq, M ).\n\nâ†’ Jq âŠ†\n\nH 1(Dq, M ) : q prime\n\n= For a ï¬�xed collection of subgroups { the particular problem under consideration, the corresponding Selmer group is\n\ndepending on\n\nJ\n\n}\n\nresâˆ’1\n\nH 1(GQ, M ).\n\nS(M ) =\n\nq (Jq)\n\nâŠ†\n\nq \\ Write H i(Q, M ) for H i(GQ, M ), and H i(Qq, M ) for H i(Dq, M ).\n\nExample. The original examples of Selmer groups come from elliptic curves. Fix an elliptic curve E and a positive integer m, and take M = E[m], the subgroup of points in E( Â¯Q) of order dividing m. There is a natural inclusion\n\nH 1(Q, E[m])\n\nE(Q)/mE(Q) Ö’\n\n(14)\n\nâ†’\n\nE( Â¯Q) is any\n\nE(Q) to the cocycle Ïƒ\n\nobtained by sending x point satisfying my = x. Similarly, for every prime q there is a natural inclusion\n\nÏƒ(y)\n\ny, where y\n\nâˆˆ\n\n7â†’\n\nâˆ’\n\nâˆˆ\n\nH 1(Qq, E[m]).\n\nE(Qq)/mE(Qq) Ö’\n\nâ†’ Deï¬�ne the Selmer group S(E[m]) in this case by taking the group Jq to be the image of E(Qq)/mE(Qq) in H 1(Qq, E[m]), for every q. This Selmer group is an important tool in studying the
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: proves that #(pT/p2\n\nR) R)\n\nT)\n\n#(\n\nâ‰¥ â‰¤\n\nâ‰¥\n\nO\n\n/Î·), then\n\nO\n\n#(pR/p2\n\nR) = #(pT/p2\n\n(13)\n\nT) = #(\n\n/Î·).\n\nO\n\nThe ï¬�rst equality in (13) shows that Ï• induces an isomorphism of tangent spaces. Wiles uses the second equality in (13) and Theorem 5.1 to deduce that T is a local\n\n16\n\nK. RUBIN AND A. SILVERBERG\n\ncomplete intersection over that\n\n(that is, there are f1, . . . , fr âˆˆ O\n\n[[x1, . . . , xr]] such\n\nO\n\nT âˆ¼=\n\n[[x1, . . . , xr]]/(f1, . . . , fr)\n\nO\n\nas morphism.\n\nalgebras). Wiles then combines these two results to prove that Ï• is an iso-\n\nO\n\n5.2. Selmer groups. In general, if M is a torsion GQ-module, a Selmer group attached to M is a subgroup of the Galois cohomology group H 1(GQ, M ) deter- mined by certain â€œlocal conditionsâ€� in the following way. If q is a prime with decomposition group Dq âŠ‚\n\nGQ, then there is a restriction map\n\nresq : H 1(GQ, M )\n\nH 1(Dq, M ).\n\nâ†’ Jq âŠ†\n\nH 1(Dq, M ) : q prime\n\n= For a ï¬�xed collection of subgroups { the particular problem under consideration, the corresponding Selmer group is\n\ndepending on\n\nJ\n\n}\n\nresâˆ’1\n\nH 1(GQ, M ).\n\nS(M ) =\n\nq (Jq)\n\nâŠ†\n\nq \\ Write H i(Q, M ) for H i(GQ, M ), and H i(Qq, M ) for H i(Dq, M ).\n\nExample. The original examples of Selmer groups come from elliptic curves. Fix an elliptic curve E and a positive integer m, and take M = E[m], the subgroup of points in E( Â¯Q) of order dividing m. There is a natural inclusion\n\nH 1(Q, E[m])\n\nE(Q)/mE(Q) Ö’\n\n(14)\n\nâ†’\n\nE( Â¯Q) is any\n\nE(Q) to the cocycle Ïƒ\n\nobtained by sending x point satisfying my = x. Similarly, for every prime q there is a natural inclusion\n\nÏƒ(y)\n\ny, where y\n\nâˆˆ\n\n7â†’\n\nâˆ’\n\nâˆˆ\n\nH 1(Qq, E[m]).\n\nE(Qq)/mE(Qq) Ö’\n\nâ†’ Deï¬�ne the Selmer group S(E[m]) in this case by taking the group Jq to be the image of E(Qq)/mE(Qq) in H 1(Qq, E[m]), for every q. This Selmer group is an important tool in studying the