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| In mathematics, the '''modularity theorem''' (formerly called the '''Taniyama–Shimura–Weil conjecture''' and several related names) states that [[elliptic curve]]s over the field of [[rational number]]s are related to [[modular form]]s. [[Andrew Wiles]] proved the modularity theorem for semistable elliptic curves, which was enough to imply [[Fermat's last theorem]]. Later, [[Christophe Breuil]], [[Brian Conrad]], [[Fred Diamond]], and [[Richard Taylor (mathematician)|Richard Taylor]] extended Wiles' techniques to prove the full modularity theorem in 2001.
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| The modularity theorem is a special case of more general conjectures due to [[Robert Langlands]]. The [[Langlands program]] seeks to attach an [[automorphic form]] or [[automorphic representation]] (a suitable generalization of a modular form) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a [[number field]]. Most cases of these extended conjectures have not yet been proved.
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| ==Statement==
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| The [[theorem]] states that any [[elliptic curve]] over '''Q''' can be obtained via a [[rational map]] with [[integer]] [[coefficient]]s from the [[classical modular curve]]
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| :<math>X_0(N)\ </math>
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| for some integer ''N''; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of level ''N''. If ''N'' is the smallest integer for which such a parametrization can be found (which by the modularity theorem itself is now known to be a number called the ''conductor''), then the parametrization may be defined in terms of a mapping generated by a particular kind of modular form of weight two and level ''N'', a normalized [[newform]] with integer ''q''-expansion, followed if need be by an [[Elliptic curve#Isogeny|isogeny]].
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| The modularity theorem implies a closely related analytic statement: to an elliptic curve ''E'' over '''Q''' we may attach a corresponding [[L-series of an elliptic curve|L-series]]. The ''L''-series is a [[Dirichlet series]], commonly written
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| :<math>L(s, E) = \sum_{n=1}^\infty \frac{a_n}{n^s}.</math>
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| The [[generating function]] of the coefficients <math>a_n</math> is then | |
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| :<math>f(q, E) = \sum_{n=1}^\infty a_n q^n.</math>
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| If we make the substitution
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| :<math>q = e^{2 \pi i \tau}\ </math>
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| we see that we have written the [[Fourier series|Fourier expansion]] of a function <math>f(\tau, E)</math> of the complex variable ''τ'', so the coefficients of the ''q''-series are also thought of as the Fourier coefficients of <math>f</math>. The function obtained in this way is, remarkably, a [[modular form|cusp form]] of weight two and level ''N'' and is also an eigenform (an eigenvector of all [[Hecke operator]]s); this is the '''Hasse–Weil conjecture''', which follows from the modularity theorem.
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| Some modular forms of weight two, in turn, correspond to [[holomorphic differential]]s for an elliptic curve. The Jacobian of the modular curve can (up to isogeny) be written as a product of irreducible [[Abelian varieties]], corresponding to Hecke eigenforms of weight 2. The 1-dimensional factors are elliptic curves (there can also be higher dimensional factors, so not all Hecke eigenforms correspond to rational elliptic curves). The curve obtained by finding the corresponding cusp form, and then constructing a curve from it, is [[Elliptic curve#Isogeny|isogenous]] to the original curve (but not, in general, isomorphic to it).
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| ==History==
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| {{harvs|txt|authorlink=Yutaka Taniyama|last=Taniyama|year=1956}} stated a preliminary (slightly incorrect) version of the conjecture at the 1955 international symposium on algebraic number theory in Tokyo and Nikko. [[Goro Shimura]] and Taniyama worked on improving its rigor until 1957. {{harvs|txt|authorlink=André Weil|last=Weil|year= 1967}} rediscovered the conjecture, and showed that it would follow from the (conjectured) functional equations for some twisted L-series of the elliptic curve; this was the first serious evidence that the conjecture might be true. Weil also showed that the conductor of the elliptic curve should be the level of the corresponding modular form. The "astounding"<ref name="Singh">''[[Fermat's Last Theorem (book)|Fermat's Last Theorem]]'', [[Simon Singh]], 1997, ISBN 1-85702-521-0</ref>{{rp|211}} conjecture (at the time known as the Taniyama–Shimura-Weil conjecture) became a part of the [[Langlands program]], a list of important conjectures needing proof or disproof.<ref name="Singh" />{{rp|211–215}}
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| The conjecture attracted considerable interest when {{harvs|txt|authorlink=Gerhard Frey|last=Frey|year=1986}} suggested that the Taniyama–Shimura–Weil conjecture implies [[Fermat's Last Theorem]]. He did this by attempting to show that any counterexample to Fermat's Last Theorem would imply the existence of at least one non-modular elliptic curve. This argument was completed when {{harvs|txt|authorlink=Jean-Pierre Serre|last=Serre|year=1987}} identified a missing link (now known as the [[epsilon conjecture]] or [[Ribet's theorem]]) in Frey's original work, followed two years later by {{harvs|txt|authorlink=Ken Ribet|last=Ribet|year=1990}}'s completion of a proof of the epsilon conjecture.
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| Even after gaining serious attention, the Taniyama–Shimura-Weil conjecture was seen by contemporary mathematicians as extraordinarily difficult or perhaps inaccessible to proof.<ref name="Singh" />{{rp|203–205, 223, 226}} For example, Wiles' ex-supervisor [[John H. Coates|John Coates]] states that it seemed "impossible to actually prove",<ref name="Singh" />{{rp|226}} and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible".<ref name="Singh" />{{rp|223}}
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| In 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]], and the full Taniyama–Shimura–Weil conjecture was finally proved by {{harvtxt|Diamond|1996}}, {{harvtxt|Conrad|Diamond|Taylor|1999}}, and {{harvtxt|Breuil|Conrad|Diamond|Taylor|2001}} who, building on Wiles' work, incrementally chipped away at the remaining cases until the full result was proved.
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| {{see|Fermat's Last Theorem|Wiles' proof of Fermat's Last Theorem}}
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| Once fully proven, the conjecture became known as the modularity theorem.
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| Several theorems in number theory{{which|date=May 2013}} similar to Fermat's Last Theorem follow from the modularity theorem. For example: no cube can be written as a sum of two [[coprime]] ''n''-th powers, ''n'' ≥ 3. (The case ''n'' = 3 was already known by [[Euler]].)
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| == References ==
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| <references />
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| *{{Citation | last1=Breuil | first1=Christophe | last2=Conrad | first2=Brian | last3=Diamond | first3=Fred | last4=Taylor | first4=Richard | title=On the modularity of elliptic curves over '''Q''': wild 3-adic exercises | doi=10.1090/S0894-0347-01-00370-8 | mr=1839918 | year=2001 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | volume=14 | issue=4 | pages=843–939}}
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| *{{Citation | last1=Conrad | first1=Brian | last2=Diamond | first2=Fred | last3=Taylor | first3=Richard | title=Modularity of certain potentially Barsotti-Tate Galois representations | doi=10.1090/S0894-0347-99-00287-8 | mr=1639612 | year=1999 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | volume=12 | issue=2 | pages=521–567}}
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| *{{Citation | editor1-last=Cornell | editor1-first=Gary | editor2-last=Silverman | editor2-first=Joseph H. | editor2-link=Joseph H. Silverman | editor3-last=Stevens | editor3-first=Glenn | title=Modular forms and Fermat's last theorem | url=http://books.google.com/books?id=Va-quzVwtMsC | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-94609-2; 978-0-387-98998-3 | mr=1638473 | year=1997}}
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| *{{Citation | last1=Darmon | first1=Henri | title=A proof of the full Shimura-Taniyama-Weil conjecture is announced | url=http://www.ams.org/notices/199911/comm-darmon.pdf | mr=1723249 | year=1999 | journal=[[Notices of the American Mathematical Society]] | issn=0002-9920 | volume=46 | issue=11 | pages=1397–1401}}Contains a gentle introduction to the theorem and an outline of the proof.
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| *{{Citation | last1=Diamond | first1=Fred | title=On deformation rings and Hecke rings | doi=10.2307/2118586 | mr=1405946 | year=1996 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=144 | issue=1 | pages=137–166}}
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| *{{Citation | last1=Frey | first1=Gerhard | title=Links between stable elliptic curves and certain Diophantine equations | mr=853387 | year=1986 | journal=Annales Universitatis Saraviensis. Series Mathematicae | issn=0933-8268 | volume=1 | issue=1 | pages=iv+40}}
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| *{{Citation | last1=Mazur | first1=Barry | author1-link=Barry Mazur | title=Number theory as gadfly | doi=10.2307/2324924 | mr=1121312 | year=1991 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 | volume=98 | issue=7 | pages=593–610}} Discusses the Taniyama-Shimura-Weil conjecture 3 years before it was proven for infinitely many cases.
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| *{{Citation | last1=Ribet | first1=Kenneth A. | title=On modular representations of Gal({{overline|Q}}/Q) arising from modular forms | doi=10.1007/BF01231195 | mr=1047143 | year=1990 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=100 | issue=2 | pages=431–476}}
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| *{{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Sur les représentations modulaires de degré 2 de Gal({{overline|Q}}/Q) | doi=10.1215/S0012-7094-87-05413-5 | mr=885783 | year=1987 | journal=[[Duke Mathematical Journal]] | issn=0012-7094 | volume=54 | issue=1 | pages=179–230}}
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| *{{Citation | last1=Shimura | first1=Goro | title=Yutaka Taniyama and his time. Very personal recollections | doi=10.1112/blms/21.2.186 | mr=976064 | year=1989 | journal=The Bulletin of the London Mathematical Society | issn=0024-6093 | volume=21 | issue=2 | pages=186–196}}
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| *{{citation|last=Taniyama|first=Yutaka |journal=Sugaku|volume=7|page=269|year=1956|title=Problem 12|language=Japanese}} English translation in {{harv|Shimura|1989|loc=p. 194}}
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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|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=141 | issue=3 | pages=553–572}}
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| *{{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}}
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| *{{Citation | last1=Wiles | first1=Andrew | author1-link=Andrew Wiles | title=Modular elliptic curves and Fermat's last theorem | jstor=2118559 | mr=1333035 | year=1995 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=141 | issue=3 | pages=443–551}}
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| *{{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=1995 | chapter=Modular forms, elliptic curves, and Fermat's last theorem | pages=243–245}}
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| ==External links==
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| *{{eom|id=S/s120140|title=Shimura–Taniyama conjecture|first=H. |last=Darmon}}
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| * {{MathWorld | urlname=Taniyama-ShimuraConjecture | title= Taniyama-Shimura Conjecture }}
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| [[Category:Algebraic curves]]
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| [[Category:Riemann surfaces]]
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| [[Category:Modular forms]]
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| [[Category:Theorems in number theory]]
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| [[Category:Theorems in algebraic geometry]]
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