second deï¬�nition of modularity for an elliptic curve.\n\nDeï¬�nition. An elliptic curve E over Q is modular if there exists an eigenform\n\nâˆ� n=1 ane2Ï€inz such that for all but ï¬�nitely many primes q,\n\n#(E(Fq)).\n\n(5) P\n\naq = q + 1\n\nâˆ’ 2. An overview\n\nThe ï¬‚ow chart shows how Fermatâ€™s Last Theorem would follow if one knew the Semistable Modular Lifting Conjecture (Conjecture 2.1) for the primes 3 and 5. 1 we discussed the upper arrow, i.e., the implication â€œSemistable Taniyama- In Â§ Fermatâ€™s Last Theoremâ€�. In this section we will discuss the Shimura Conjecture other implications in the ï¬‚ow chart. The implication given by the lowest arrow is straightforward (Proposition 2.3), while the middle one uses an ingenious idea of Wiles (Proposition 2.4).\n\nâ‡’\n\nFermatâ€™s Last Theorem\n\nâœ»\n\nSemistable Taniyama-Shimura Conjecture\n\nâœ»\n\n(cid:0)\n\nâ�… â�…\n\n(cid:0)\n\nSemistable Taniyama-Shimura for Â¯Ï�E,3 irreducible\n\nSemistable Modular Lifting for p = 5\n\nâœ»\n\n(cid:0) (cid:0)\n\nâ�…\n\nâ�…\n\nSemistable Modular Lifting for p = 3\n\nLanglands-Tunnell Theorem\n\nSemistable Modular Lifting Conjecture\n\nFermatâ€™s Last Theorem .\n\nâ‡’\n\nRemark . By the Modular Lifting Conjecture we will mean the Semistable Modular Lifting Conjecture with the hypothesis of semistability removed. The arguments of this section can also be used to show that the Modular Lifting Conjecture for p = 3 and 5, together with the Langlands-Tunnell Theorem, imply the full Taniyama- Shimura Conjecture.\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n7\n\n2.1. Semistable Modular Lifting. Let Â¯Q denote the algebraic closure of Q in C, and let GQ be the Galois group Gal( Â¯Q/Q). If p is a prime, write\n\nFÃ— p\n\nÂ¯Îµp : GQ\n\nâ†’\n\nfor the character giving the action of GQ on the p-th roots of unity. For the facts about elliptic curves stated below, see [39]. If E is an elliptic curve over Q and F is a subï¬�eld of the complex numbers, there is a natural
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: second deï¬�nition of modularity for an elliptic curve.\n\nDeï¬�nition. An elliptic curve E over Q is modular if there exists an eigenform\n\nâˆ� n=1 ane2Ï€inz such that for all but ï¬�nitely many primes q,\n\n#(E(Fq)).\n\n(5) P\n\naq = q + 1\n\nâˆ’ 2. An overview\n\nThe ï¬‚ow chart shows how Fermatâ€™s Last Theorem would follow if one knew the Semistable Modular Lifting Conjecture (Conjecture 2.1) for the primes 3 and 5. 1 we discussed the upper arrow, i.e., the implication â€œSemistable Taniyama- In Â§ Fermatâ€™s Last Theoremâ€�. In this section we will discuss the Shimura Conjecture other implications in the ï¬‚ow chart. The implication given by the lowest arrow is straightforward (Proposition 2.3), while the middle one uses an ingenious idea of Wiles (Proposition 2.4).\n\nâ‡’\n\nFermatâ€™s Last Theorem\n\nâœ»\n\nSemistable Taniyama-Shimura Conjecture\n\nâœ»\n\n(cid:0)\n\nâ�… â�…\n\n(cid:0)\n\nSemistable Taniyama-Shimura for Â¯Ï�E,3 irreducible\n\nSemistable Modular Lifting for p = 5\n\nâœ»\n\n(cid:0) (cid:0)\n\nâ�…\n\nâ�…\n\nSemistable Modular Lifting for p = 3\n\nLanglands-Tunnell Theorem\n\nSemistable Modular Lifting Conjecture\n\nFermatâ€™s Last Theorem .\n\nâ‡’\n\nRemark . By the Modular Lifting Conjecture we will mean the Semistable Modular Lifting Conjecture with the hypothesis of semistability removed. The arguments of this section can also be used to show that the Modular Lifting Conjecture for p = 3 and 5, together with the Langlands-Tunnell Theorem, imply the full Taniyama- Shimura Conjecture.\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n7\n\n2.1. Semistable Modular Lifting. Let Â¯Q denote the algebraic closure of Q in C, and let GQ be the Galois group Gal( Â¯Q/Q). If p is a prime, write\n\nFÃ— p\n\nÂ¯Îµp : GQ\n\nâ†’\n\nfor the character giving the action of GQ on the p-th roots of unity. For the facts about elliptic curves stated below, see [39]. If E is an elliptic curve over Q and F is a subï¬�eld of the complex numbers, there is a natural