given by the local conditions\n\nâˆ— =\n\nJ âˆ— q âŠ†\n\n, where\n\nJ\n\n{\n\nthe orthogonal complement of Jq under H 1(Qq, Ë†M )\n\n,\n\nif q / if q\n\n, âˆˆ L . âˆˆ L\n\niq\n\nJ âˆ— q =\n\nh\n\n(\n\nH 1(Q, Ë†M ), deï¬�ne\n\nIf d\n\nâˆˆ\n\nZ/mZ\n\nÎ¸d :\n\nJq â†’\n\nYqâˆˆL\n\nby\n\nÎ¸d((cq)) =\n\ncq, resq(d) h\n\niq.\n\nXqâˆˆL\n\nWrite resL : H 1(Q, M ) maps. By (15) and the deï¬�nition of J âˆ— in addition resL is injective on S(M ), then\n\nqâˆˆL H 1(Qq, M ) for the product of the restriction ker(Î¸d). If\n\nâ†’\n\nSâˆ—\n\nq , if d\n\nL, then resL(S(M ))\n\nâˆˆ\n\nâŠ†\n\nQ\n\n#(S(M ))\n\n#\n\nker(Î¸d)\n\n.\n\nâ‰¤\n\n(cid:0) \\dâˆˆSâˆ—\n\nL\n\n(cid:1)\n\nThe diï¬ƒculty is to produce enough cohomology classes in Sâˆ—\n\nL to show that the right side of the above inequality is small. Following Kolyvagin, an Euler system is Sâˆ— L for a large (inï¬�nite) collection of sets of a compatible collection of classes Îº( )) primes is related to resâ„“(Îº( )). Once an Euler system is given, Kolyvagin has an inductive procedure for choosing a set\n\n) L\n\nâˆˆ\n\n. Loosely speaking, compatible means that if â„“ /\n\n, then resâ„“(Îº(\n\nâ„“\n\nL\n\nâˆˆ L\n\nL âˆª {\n\n}\n\nL\n\nsuch that\n\nL\n\nresL is injective on S(M ),\n\nâ€¢\n\nPâŠ†L ker(Î¸Îº(P)) can be computed in terms of Îº( âˆ…\n\n).\n\nT\n\n18\n\nK. RUBIN AND A. SILVERBERG\n\nSâˆ—\n\nSâˆ—\n\n, then Sâˆ—\n\nL.)\n\nL, so Îº(\n\n)\n\n(Note that if\n\nP âŠ†\n\nP For several important Selmer groups it is possible to construct Euler systems for\n\nâˆˆ\n\nP âŠ† L\n\nwhich Kolyvaginâ€™s procedure produces a set\n\nactually giving an equality\n\nL ker(Î¸Îº(P))\n\n#(S(M )) = #\n\n.\n\n(cid:0) \\PâŠ†L This is what Wiles needs to do for the Selmer group SD(Vn). There are several examples in the literature where this kind of argument is worked out in some detail. For the simplest case, where the Selmer group in question is the ideal class group ) are constructed from cyclotomic units, of a real abelian number ï¬�eld and the Îº(
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. ->: given by the local conditions\n\nâˆ— =\n\nJ âˆ— q âŠ†\n\n, where\n\nJ\n\n{\n\nthe orthogonal complement of Jq under H 1(Qq, Ë†M )\n\n,\n\nif q / if q\n\n, âˆˆ L . âˆˆ L\n\niq\n\nJ âˆ— q =\n\nh\n\n(\n\nH 1(Q, Ë†M ), deï¬�ne\n\nIf d\n\nâˆˆ\n\nZ/mZ\n\nÎ¸d :\n\nJq â†’\n\nYqâˆˆL\n\nby\n\nÎ¸d((cq)) =\n\ncq, resq(d) h\n\niq.\n\nXqâˆˆL\n\nWrite resL : H 1(Q, M ) maps. By (15) and the deï¬�nition of J âˆ— in addition resL is injective on S(M ), then\n\nqâˆˆL H 1(Qq, M ) for the product of the restriction ker(Î¸d). If\n\nâ†’\n\nSâˆ—\n\nq , if d\n\nL, then resL(S(M ))\n\nâˆˆ\n\nâŠ†\n\nQ\n\n#(S(M ))\n\n#\n\nker(Î¸d)\n\n.\n\nâ‰¤\n\n(cid:0) \\dâˆˆSâˆ—\n\nL\n\n(cid:1)\n\nThe diï¬ƒculty is to produce enough cohomology classes in Sâˆ—\n\nL to show that the right side of the above inequality is small. Following Kolyvagin, an Euler system is Sâˆ— L for a large (inï¬�nite) collection of sets of a compatible collection of classes Îº( )) primes is related to resâ„“(Îº( )). Once an Euler system is given, Kolyvagin has an inductive procedure for choosing a set\n\n) L\n\nâˆˆ\n\n. Loosely speaking, compatible means that if â„“ /\n\n, then resâ„“(Îº(\n\nâ„“\n\nL\n\nâˆˆ L\n\nL âˆª {\n\n}\n\nL\n\nsuch that\n\nL\n\nresL is injective on S(M ),\n\nâ€¢\n\nPâŠ†L ker(Î¸Îº(P)) can be computed in terms of Îº( âˆ…\n\n).\n\nT\n\n18\n\nK. RUBIN AND A. SILVERBERG\n\nSâˆ—\n\nSâˆ—\n\n, then Sâˆ—\n\nL.)\n\nL, so Îº(\n\n)\n\n(Note that if\n\nP âŠ†\n\nP For several important Selmer groups it is possible to construct Euler systems for\n\nâˆˆ\n\nP âŠ† L\n\nwhich Kolyvaginâ€™s procedure produces a set\n\nactually giving an equality\n\nL ker(Î¸Îº(P))\n\n#(S(M )) = #\n\n.\n\n(cid:0) \\PâŠ†L This is what Wiles needs to do for the Selmer group SD(Vn). There are several examples in the literature where this kind of argument is worked out in some detail. For the simplest case, where the Selmer group in question is the ideal class group ) are constructed from cyclotomic units, of a real abelian number ï¬�eld and the Îº(