(see II.4.D of [24] and appendix of [22]). In some cases the equivalence is a deep result. For Wilesâ€™ 1.7 proof of Fermatâ€™s Last Theorem it suï¬ƒces to use only the deï¬�nition given in below.\n\nÂ§\n\nÂ§\n\n1.5. Semistability.\n\nDeï¬�nition. An elliptic curve over Q is semistable at the prime q if it is isomorphic to an elliptic curve over Q which modulo q either is nonsingular or has a singu- lar point with two distinct tangent directions. An elliptic curve over Q is called semistable if it is semistable at every prime.\n\nExample. The elliptic curve y2 = x(x + 32)(x isomorphic to y2 + xy + y = x3 + x2 x(x + 42)(x\n\n42) is semistable because it is âˆ’ 10, but the elliptic curve y2 =\n\n10x\n\nâˆ’\n\nâˆ’\n\n32) is not semistable (it is not semistable at 2).\n\nâˆ’\n\n2 we explain how Wiles shows that his main result on Galois representations (Theorem 5.3) implies the following part of the Taniyama-Shimura Conjecture.\n\nBeginning in\n\nÂ§\n\nSemistable Taniyama-Shimura Conjecture. Every semistable elliptic curve over Q is modular.\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n5\n\nProposition 1.2. The Semistable Taniyama-Shimura Conjecture implies Theorem 1.1.\n\nProof. If A and B are distinct, nonzero, relatively prime integers, write EA,B for the elliptic curve deï¬�ned by y2 = x(x + A)(x + B). Since EA,B and Eâˆ’A,âˆ’B are isomorphic over the complex numbers (i.e., as Riemann surfaces), EA,B is modular if and only if Eâˆ’A,âˆ’B is modular. If further AB(A B) is divisible by 16, then either EA,B or Eâˆ’A,âˆ’B is semistable (this is easy to check directly; see for example I.1 of [24]). The Semistable Taniyama-Shimura Conjecture now implies that both Â§ EA,B and Eâˆ’A,âˆ’B are modular, and thus implies Theorem 1.1.\n\nâˆ’\n\nRemark . In 1.1 we saw that Theorem 1.1 and Ribetâ€™s Theorem together imply Fermatâ€™s Last Theorem. Therefore, the Semistable Taniyama-Shimura Conjecture implies Fermatâ€™s Last Theorem.\n\nÂ§\n\n1.6. Modular forms. In this paper we
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: (see II.4.D of [24] and appendix of [22]). In some cases the equivalence is a deep result. For Wilesâ€™ 1.7 proof of Fermatâ€™s Last Theorem it suï¬ƒces to use only the deï¬�nition given in below.\n\nÂ§\n\nÂ§\n\n1.5. Semistability.\n\nDeï¬�nition. An elliptic curve over Q is semistable at the prime q if it is isomorphic to an elliptic curve over Q which modulo q either is nonsingular or has a singu- lar point with two distinct tangent directions. An elliptic curve over Q is called semistable if it is semistable at every prime.\n\nExample. The elliptic curve y2 = x(x + 32)(x isomorphic to y2 + xy + y = x3 + x2 x(x + 42)(x\n\n42) is semistable because it is âˆ’ 10, but the elliptic curve y2 =\n\n10x\n\nâˆ’\n\nâˆ’\n\n32) is not semistable (it is not semistable at 2).\n\nâˆ’\n\n2 we explain how Wiles shows that his main result on Galois representations (Theorem 5.3) implies the following part of the Taniyama-Shimura Conjecture.\n\nBeginning in\n\nÂ§\n\nSemistable Taniyama-Shimura Conjecture. Every semistable elliptic curve over Q is modular.\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n5\n\nProposition 1.2. The Semistable Taniyama-Shimura Conjecture implies Theorem 1.1.\n\nProof. If A and B are distinct, nonzero, relatively prime integers, write EA,B for the elliptic curve deï¬�ned by y2 = x(x + A)(x + B). Since EA,B and Eâˆ’A,âˆ’B are isomorphic over the complex numbers (i.e., as Riemann surfaces), EA,B is modular if and only if Eâˆ’A,âˆ’B is modular. If further AB(A B) is divisible by 16, then either EA,B or Eâˆ’A,âˆ’B is semistable (this is easy to check directly; see for example I.1 of [24]). The Semistable Taniyama-Shimura Conjecture now implies that both Â§ EA,B and Eâˆ’A,âˆ’B are modular, and thus implies Theorem 1.1.\n\nâˆ’\n\nRemark . In 1.1 we saw that Theorem 1.1 and Ribetâ€™s Theorem together imply Fermatâ€™s Last Theorem. Therefore, the Semistable Taniyama-Shimura Conjecture implies Fermatâ€™s Last Theorem.\n\nÂ§\n\n1.6. Modular forms. In this paper we