bound for the Selmer group in the semistable caseâ€� (see 5.4 below) â€œis not yet Â§ complete as it stands,â€� but that he believes he will be able to ï¬�nish it in the near future using the ideas explained in his Cambridge lectures. While Wilesâ€™ proof of Theorem 5.3 below and Fermatâ€™s Last Theorem depends on the calculation he referred to in his December announcement, Theorem 5.4 and Corollary 5.5 do not. Wilesâ€™ work provides for the ï¬�rst time inï¬�nitely many modular elliptic curves over the rational numbers which are not isomorphic over the complex numbers (see 5.5 for an explicit inï¬�nite family).\n\n5.3 and\n\nÂ§\n\nÂ§\n\nNotation. The integers, rational numbers, complex numbers, and p-adic integers will be denoted Z, Q, C, and Zp, respectively. If F is a ï¬�eld, then Â¯F denotes an algebraic closure of F .\n\n1. Connection between Fermatâ€™s Last Theorem and elliptic curves\n\n1.1. Fermatâ€™s Last Theorem follows from modularity of elliptic curves. Suppose Fermatâ€™s Last Theorem were false. Then there would exist nonzero integers a, b, c, and n > 2 such that an + bn = cn. It is easy to see that no generality is lost by assuming that n is a prime greater than three (or greater than four million, by [2]; see [14] for n = 3 and 4) and that a and b are relatively prime. Write down the cubic curve:\n\ny2 = x(x + an)(x\n\nbn).\n\n(1)\n\nâˆ’\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n3\n\n1.4 we will explain what it means for an elliptic curve to be modular. Kenneth Ribet [27] proved that if n is a prime greater than three, a, b, and c are nonzero integers, and an + bn = cn, then the elliptic curve (1) is not modular. But the results announced by Wiles imply the following.\n\nIn\n\n1.3 we will see that such curves are elliptic curves, and in\n\nÂ§\n\nÂ§\n\nTheorem 1.1 (Wiles). If A and B are distinct, nonzero, relatively prime integers, and AB(A\n\nB) is divisible by 16, then the elliptic curve\n\nâˆ’\n\ny2 = x(x + A)(x + B)\n\nis modular.\n\nbn with
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: bound for the Selmer group in the semistable caseâ€� (see 5.4 below) â€œis not yet Â§ complete as it stands,â€� but that he believes he will be able to ï¬�nish it in the near future using the ideas explained in his Cambridge lectures. While Wilesâ€™ proof of Theorem 5.3 below and Fermatâ€™s Last Theorem depends on the calculation he referred to in his December announcement, Theorem 5.4 and Corollary 5.5 do not. Wilesâ€™ work provides for the ï¬�rst time inï¬�nitely many modular elliptic curves over the rational numbers which are not isomorphic over the complex numbers (see 5.5 for an explicit inï¬�nite family).\n\n5.3 and\n\nÂ§\n\nÂ§\n\nNotation. The integers, rational numbers, complex numbers, and p-adic integers will be denoted Z, Q, C, and Zp, respectively. If F is a ï¬�eld, then Â¯F denotes an algebraic closure of F .\n\n1. Connection between Fermatâ€™s Last Theorem and elliptic curves\n\n1.1. Fermatâ€™s Last Theorem follows from modularity of elliptic curves. Suppose Fermatâ€™s Last Theorem were false. Then there would exist nonzero integers a, b, c, and n > 2 such that an + bn = cn. It is easy to see that no generality is lost by assuming that n is a prime greater than three (or greater than four million, by [2]; see [14] for n = 3 and 4) and that a and b are relatively prime. Write down the cubic curve:\n\ny2 = x(x + an)(x\n\nbn).\n\n(1)\n\nâˆ’\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n3\n\n1.4 we will explain what it means for an elliptic curve to be modular. Kenneth Ribet [27] proved that if n is a prime greater than three, a, b, and c are nonzero integers, and an + bn = cn, then the elliptic curve (1) is not modular. But the results announced by Wiles imply the following.\n\nIn\n\n1.3 we will see that such curves are elliptic curves, and in\n\nÂ§\n\nÂ§\n\nTheorem 1.1 (Wiles). If A and B are distinct, nonzero, relatively prime integers, and AB(A\n\nB) is divisible by 16, then the elliptic curve\n\nâˆ’\n\ny2 = x(x + A)(x + B)\n\nis modular.\n\nbn with