Modularity theorem: Difference between revisions

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Even after gaining serious attention, the Taniyama–Shimura–Weil conjecture was seen by contemporary mathematicians as extraordinarily difficult to prove or perhaps even inaccessible to proof<!--{{harv|Singh|1997|pp=203–205, 223, 226}}-->.{{sfn|Singh|1997|pp=203–205, 223, 226}} For example, Wiles's Ph.D. supervisor [[John H. Coates|John Coates]] states that it seemed "impossible to actually prove", and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible".

Andrew Wiles,{{sfn|Wiles|1995}}<!--{{harvs|txt|authorlink=Andrew Wiles|last=Wiles|year=1995}}--> with some help from [[Richard Taylor (mathematician)|Richard Taylor]], proved the Taniyama–Shimura–Weil conjecture for all [[semistable elliptic curve]]s, which he used to prove Fermat's Last Theorem,{{sfnm|Wiles|1995a|Wiles|1995b}}<!--{{harvs|txt|authorlink=Andrew Wiles|last=Wiles|year=1995}}--> and the full Taniyama–Shimura–Weil conjecture was finally proved by Diamond,{{sfn|Diamond|1996}}<!--{{harvtxt|Diamond|1996}}--> Conrad, Diamond & Taylor; and Breuil, Conrad, Diamond & Taylor; building on Wiles's work, incrementally chipped away at the remaining cases until the full result was proved.{{sfn|Conrad|Diamond|Taylor|1999}}<!--{{harvtxt|Conrad|Diamond|Taylor|1999}}--> and Breuil, Conrad, Diamond & Taylor{{sfn|Breuil|Conrad|Diamond|Taylor|2001}}<!--{{harvtxt|Breuil|Conrad|Diamond|Taylor|2001}}--> who, building on Wiles's work, incrementally chipped away at the remaining cases until the full result was proved.

{{further|Fermat's Last Theorem|Wiles's proof of Fermat's Last Theorem}}

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==Notes==

{{reflist|20em}}

== References ==

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*{{Citation | last1=Taylor | first1=Richard | last2=Wiles | first2=Andrew | author2-link=Andrew Wiles | title=Ring-theoretic properties of certain Hecke algebras | doi=10.2307/2118560 | mr=1333036 | year=1995 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=141 | issue=3 | pages=553–572| jstor=2118560 | citeseerx=10.1.1.128.531 }}

*{{Citation | last1=Weil | first1=André | author1-link=André Weil | title=Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen | doi=10.1007/BF01361551 | mr=0207658 | year=1967 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=168 | pages=149–156}}

*{{Citation | last1=Wiles | first1=Andrew | author1-link=Andrew Wiles | title=Modular elliptic curves and Fermat's last theorem | jstor=2118559 | mr=1333035 | year=19951995a | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=141 | issue=3 | pages=443–551 | doi=10.2307/2118559| citeseerx=10.1.1.169.9076 }}

*{{Citation | last1=Wiles | first1=Andrew | author1-link=Andrew Wiles | title=Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) | publisher=Birkhäuser | ___location=Basel, Boston, Berlin | mr=1403925 | year=19951995b | chapter=Modular forms, elliptic curves, and Fermat's last theorem | pages=243–245}}

==External links==