of a real abelian number ï¬�eld and the Îº( L see [29]. For other cases involving ideal class groups and Selmer groups of elliptic curves, see [19], [31], [30], [13].\n\n(cid:1)\n\n5.4. Wilesâ€™ geometric Euler system. The task now is to construct an Euler system of cohomology classes with which to bound #(SD(Vn)) using Kolyvaginâ€™s method. This is the most technically diï¬ƒcult part of Wilesâ€™ proof and is the part of Wilesâ€™ work he referred to as not yet complete in his December announcement. We give only general remarks about Wilesâ€™ construction.\n\nThe ï¬�rst step in the construction is due to Flach [10]. He constructed classes consisting of just one prime. This allows one to bound the ) L\n\nSâˆ—\n\nÎº( exponent of SD(Vn), but not its order.\n\nL for sets\n\nâˆˆ\n\nL\n\nEvery Euler system starts with some explicit, concrete objects. Earlier examples of Euler systems come from cyclotomic or elliptic units, Gauss sums, or Heegner points on elliptic curves. Wiles (following Flach) constructs his cohomology classes from modular units, i.e., meromorphic functions on modular curves which are holo- morphic and nonzero away from the cusps. More precisely, Îº( ) comes from an explicit function on the modular curve X1(L, N ), the curve obtained by taking the quotient space of the upper half plane by the action of the group\n\nL\n\na b c d\n\nSL2(Z) : c\n\n1 (mod L) } â‰¡ â„“âˆˆL â„“ and where N is the N of (T2) of\n\n0\n\n(mod LN ),\n\na\n\nd\n\n,\n\nÎ“1(L, N ) =\n\nâˆˆ\n\nâ‰¡\n\nâ‰¡\n\n{ (cid:1) (cid:0) and adjoining the cusps, where L = The construction and study of the classes Îº( [8], [9] and others.\n\n4.2. ) rely heavily on results of Faltings\n\nÂ§\n\nL\n\nQ\n\n5.5. Wilesâ€™ results. Wiles announced two main results (Theorems 5.3 and 5.4 below) in the direction of Mazurâ€™s conjecture, under two diï¬€erent sets of hypotheses on the representation Â¯Ï�. Theorem 5.3 implies the Semistable Taniyama-Shimura Conjecture and Fermatâ€™s Last Theorem. Wilesâ€™ proof
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. ->: of a real abelian number ï¬�eld and the Îº( L see [29]. For other cases involving ideal class groups and Selmer groups of elliptic curves, see [19], [31], [30], [13].\n\n(cid:1)\n\n5.4. Wilesâ€™ geometric Euler system. The task now is to construct an Euler system of cohomology classes with which to bound #(SD(Vn)) using Kolyvaginâ€™s method. This is the most technically diï¬ƒcult part of Wilesâ€™ proof and is the part of Wilesâ€™ work he referred to as not yet complete in his December announcement. We give only general remarks about Wilesâ€™ construction.\n\nThe ï¬�rst step in the construction is due to Flach [10]. He constructed classes consisting of just one prime. This allows one to bound the ) L\n\nSâˆ—\n\nÎº( exponent of SD(Vn), but not its order.\n\nL for sets\n\nâˆˆ\n\nL\n\nEvery Euler system starts with some explicit, concrete objects. Earlier examples of Euler systems come from cyclotomic or elliptic units, Gauss sums, or Heegner points on elliptic curves. Wiles (following Flach) constructs his cohomology classes from modular units, i.e., meromorphic functions on modular curves which are holo- morphic and nonzero away from the cusps. More precisely, Îº( ) comes from an explicit function on the modular curve X1(L, N ), the curve obtained by taking the quotient space of the upper half plane by the action of the group\n\nL\n\na b c d\n\nSL2(Z) : c\n\n1 (mod L) } â‰¡ â„“âˆˆL â„“ and where N is the N of (T2) of\n\n0\n\n(mod LN ),\n\na\n\nd\n\n,\n\nÎ“1(L, N ) =\n\nâˆˆ\n\nâ‰¡\n\nâ‰¡\n\n{ (cid:1) (cid:0) and adjoining the cusps, where L = The construction and study of the classes Îº( [8], [9] and others.\n\n4.2. ) rely heavily on results of Faltings\n\nÂ§\n\nL\n\nQ\n\n5.5. Wilesâ€™ results. Wiles announced two main results (Theorems 5.3 and 5.4 below) in the direction of Mazurâ€™s conjecture, under two diï¬€erent sets of hypotheses on the representation Â¯Ï�. Theorem 5.3 implies the Semistable Taniyama-Shimura Conjecture and Fermatâ€™s Last Theorem. Wilesâ€™ proof