= x(x + A)(x + B)\n\nis modular.\n\nbn with a, b, c, and n coming from our hypothetical solution to a Fermat equation as above, we see that the conditions of Theorem 1.1 are satisï¬�ed since n 5 and one of a, b, and c is even. Thus Theorem 1.1 and Ribetâ€™s result together imply Fermatâ€™s Last Theorem!\n\nTaking A = an and B =\n\nâˆ’\n\nâ‰¥\n\n1.2. History. The story of the connection between Fermatâ€™s Last Theorem and elliptic curves begins in 1955, when Yutaka Taniyama (1927â€“1958) posed problems which may be viewed as a weaker version of the following conjecture (see [38]).\n\nTaniyama-Shimura Conjecture. Every elliptic curve over Q is modular.\n\nThe conjecture in the present form was made by Goro Shimura around 1962â€“64 and has become better understood due to work of Shimura [33â€“37] and of AndrÂ´e Weil [42] (see also [7]). The Taniyama-Shimura Conjecture is one of the major conjectures in number theory.\n\nBeginning in the late 1960s [15â€“18], Yves Hellegouarch connected Fermat equa- tions an + bn = cn with elliptic curves of the form (1) and used results about Fer- matâ€™s Last Theorem to prove results about elliptic curves. The landscape changed abruptly in 1985 when Gerhard Frey stated in a lecture at Oberwolfach that elliptic curves arising from counterexamples to Fermatâ€™s Last Theorem could not be mod- ular [11]. Shortly thereafter Ribet [27] proved this, following ideas of Jean-Pierre Serre [32] (see [24] for a survey). In other words, â€œTaniyama-Shimura Conjecture\n\nFermatâ€™s Last Theoremâ€�. Thus, the stage was set. A proof of the Taniyama-Shimura Conjecture (or enough of it to know that elliptic curves coming from Fermat equations are modular) would be a proof of Fermatâ€™s Last Theorem.\n\nâ‡’\n\n1.3. Elliptic curves.\n\nDeï¬�nition. An elliptic curve over Q is a nonsingular curve deï¬�ned by an equation of the form\n\ny2 + a1xy + a3y = x3 + a2x2 + a4x + a6\n\n(2)\n\nwhere the coeï¬ƒcients ai are integers. The solution ( on the elliptic
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: = x(x + A)(x + B)\n\nis modular.\n\nbn with a, b, c, and n coming from our hypothetical solution to a Fermat equation as above, we see that the conditions of Theorem 1.1 are satisï¬�ed since n 5 and one of a, b, and c is even. Thus Theorem 1.1 and Ribetâ€™s result together imply Fermatâ€™s Last Theorem!\n\nTaking A = an and B =\n\nâˆ’\n\nâ‰¥\n\n1.2. History. The story of the connection between Fermatâ€™s Last Theorem and elliptic curves begins in 1955, when Yutaka Taniyama (1927â€“1958) posed problems which may be viewed as a weaker version of the following conjecture (see [38]).\n\nTaniyama-Shimura Conjecture. Every elliptic curve over Q is modular.\n\nThe conjecture in the present form was made by Goro Shimura around 1962â€“64 and has become better understood due to work of Shimura [33â€“37] and of AndrÂ´e Weil [42] (see also [7]). The Taniyama-Shimura Conjecture is one of the major conjectures in number theory.\n\nBeginning in the late 1960s [15â€“18], Yves Hellegouarch connected Fermat equa- tions an + bn = cn with elliptic curves of the form (1) and used results about Fer- matâ€™s Last Theorem to prove results about elliptic curves. The landscape changed abruptly in 1985 when Gerhard Frey stated in a lecture at Oberwolfach that elliptic curves arising from counterexamples to Fermatâ€™s Last Theorem could not be mod- ular [11]. Shortly thereafter Ribet [27] proved this, following ideas of Jean-Pierre Serre [32] (see [24] for a survey). In other words, â€œTaniyama-Shimura Conjecture\n\nFermatâ€™s Last Theoremâ€�. Thus, the stage was set. A proof of the Taniyama-Shimura Conjecture (or enough of it to know that elliptic curves coming from Fermat equations are modular) would be a proof of Fermatâ€™s Last Theorem.\n\nâ‡’\n\n1.3. Elliptic curves.\n\nDeï¬�nition. An elliptic curve over Q is a nonsingular curve deï¬�ned by an equation of the form\n\ny2 + a1xy + a3y = x3 + a2x2 + a4x + a6\n\n(2)\n\nwhere the coeï¬ƒcients ai are integers. The solution ( on the elliptic