RUBIN AND A. SILVERBERG\n\nhistory of Fermatâ€™s Last Theorem (last because it is the last of Fermatâ€™s questions to be answered) see [5], [6], and [26].\n\nWhat Andrew Wiles announced in Cambridge was that he could prove â€œmanyâ€� elliptic curves are modular, suï¬ƒciently many to imply Fermatâ€™s Last Theorem. In this paper we will explain Wilesâ€™ work on elliptic curves and its connection with 1 we introduce elliptic curves and modularity, and Fermatâ€™s Last Theorem. give the connection between Fermatâ€™s Last Theorem and the Taniyama-Shimura Conjecture on the modularity of elliptic curves. In 2 we describe how Wiles re- duces the proof of the Taniyama-Shimura Conjecture to what we call the Modular Lifting Conjecture (which can be viewed as a weak form of the Taniyama-Shimura Conjecture), by using a theorem of Langlands and Tunnell. In 4 we show Â§ how the Semistable Modular Lifting Conjecture is related to a conjecture of Mazur on deformations of Galois representations (Conjecture 4.2), and in 5 we describe Wilesâ€™ method of attack on this conjecture. In order to make this survey as acces- sible as possible to nonspecialists, the more technical details are postponed as long as possible, some of them to the appendices.\n\nIn\n\nÂ§\n\nÂ§\n\n3 and Â§\n\nÂ§\n\nMuch of this report is based on Wilesâ€™ lectures in Cambridge. The authors apol- ogize for any errors we may have introduced. We also apologize to those whose mathematical contributions we, due to our incomplete understanding, do not prop- erly acknowledge.\n\nThe ideas Wiles introduced in his Cambridge lectures will have an important inï¬‚uence on research in number theory. Because of the great interest in this subject and the lack of a publicly available manuscript, we hope this report will be useful to the mathematics community. In early December 1993, shortly before this paper went to press, Wiles announced that â€œthe ï¬�nal calculation of a precise upper bound for the Selmer group in the semistable
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: RUBIN AND A. SILVERBERG\n\nhistory of Fermatâ€™s Last Theorem (last because it is the last of Fermatâ€™s questions to be answered) see [5], [6], and [26].\n\nWhat Andrew Wiles announced in Cambridge was that he could prove â€œmanyâ€� elliptic curves are modular, suï¬ƒciently many to imply Fermatâ€™s Last Theorem. In this paper we will explain Wilesâ€™ work on elliptic curves and its connection with 1 we introduce elliptic curves and modularity, and Fermatâ€™s Last Theorem. give the connection between Fermatâ€™s Last Theorem and the Taniyama-Shimura Conjecture on the modularity of elliptic curves. In 2 we describe how Wiles re- duces the proof of the Taniyama-Shimura Conjecture to what we call the Modular Lifting Conjecture (which can be viewed as a weak form of the Taniyama-Shimura Conjecture), by using a theorem of Langlands and Tunnell. In 4 we show Â§ how the Semistable Modular Lifting Conjecture is related to a conjecture of Mazur on deformations of Galois representations (Conjecture 4.2), and in 5 we describe Wilesâ€™ method of attack on this conjecture. In order to make this survey as acces- sible as possible to nonspecialists, the more technical details are postponed as long as possible, some of them to the appendices.\n\nIn\n\nÂ§\n\nÂ§\n\n3 and Â§\n\nÂ§\n\nMuch of this report is based on Wilesâ€™ lectures in Cambridge. The authors apol- ogize for any errors we may have introduced. We also apologize to those whose mathematical contributions we, due to our incomplete understanding, do not prop- erly acknowledge.\n\nThe ideas Wiles introduced in his Cambridge lectures will have an important inï¬‚uence on research in number theory. Because of the great interest in this subject and the lack of a publicly available manuscript, we hope this report will be useful to the mathematics community. In early December 1993, shortly before this paper went to press, Wiles announced that â€œthe ï¬�nal calculation of a precise upper bound for the Selmer group in the semistable