In mathematics, the modularity theorem establishes an important connection between elliptic curves over the field of rational numbers and modular forms, certain analytic functions introduced in 19th century mathematics. Andrew Wiles proved the modularity theorem for all semistable elliptic curves over the rationals, with some help from Richard Taylor, in 1995. The remaining (non-semistable) cases were subsequently settled by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor by 2001. Before its proof it was called the Taniyama–Shimura–Weil conjecture, or related names. Ribet had previously proved that the modularity theorem implied Fermat's Last Theorem. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... Modular form - Wikipedia /**/ @import /skins-1. ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ... For the French mathematician with work in the area of elliptic curves, see André Weil. ... Richard Taylor (born 19 May 1962) is a British mathematician working in the field of number theory. ... Brian Conrad (b. ... Fred Diamond (born November 19, 1964) is an American mathematician. ... Richard Taylor (born 19 May 1962) is a British mathematician working in the field of number theory. ... Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ...

The modularity theorem is a special case of more general conjectures due to Robert Langlands. In the Langlands programme, to every elliptic curve over a number field one can associate an automorphic form or automorphic representation, a suitable generalization of a modular form. Most cases of these extended conjectures have not yet been proved. Robert Langlands (born 1936 in Canada) is one of the most significant mathematicians of the 20th century, with profound insights in number theory and representation theory. ... The Langlands program is a web of far-reaching and influential conjectures that connect number theory and the representation theory of certain groups. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms. ... In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms. ...

Statement

The theorem states that any elliptic curve over Q can be obtained via a rational map with integer coefficients from the classical modular curve In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. ... The integers are commonly denoted by the above symbol. ... In mathematics, a coefficient is a constant multiplicative factor of a certain object. ... In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equation Φn(x, y)=0, where for the j-invariant j(τ), x=j(n τ), y=j(τ) is a point on the curve. ...

X₀(N)

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 isogeny. In mathematics, the Atkin-Lehner theory is an algebraic part of the theory of modular forms, in which the concept of newform is defined. ... In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ...

From another point of view, given an elliptic curve E over Q we may define a corresponding L-series. The L-series is a Dirichlet series which we may write The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary number theory. ... In mathematics, a Dirichlet series, one of a number of concepts named in honor of Johann Peter Gustav Lejeune Dirichlet, is a series of the form The most famous of Dirichlet series is which is the Riemann zeta function. ...

We can take the same coefficients, and use them to define a function in powers of q

If we make the substitution q = exp(2πiτ), then the series becomes a Fourier series, and so the coefficients are sometimes called "q-series coefficients", but other times "Fourier coefficients". The function obtained in this way, remarkably, is a cusp form of weight two and level N and is also an eigenform (an eigenvector of all Hecke operators); this is the Hasse–Weil conjecture, which follows from the modularity theorem. The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... Modular form - Wikipedia /**/ @import /skins-1. ... In mathematics, in particular in the theory of modular forms, a Hecke operator is a certain kind of averaging operator that plays a significant role in the structure of vector spaces of modular forms (and more general automorphic representations). ...

Some modular forms of level two, in turn, correspond to holomorphic differentials 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 we obtain by finding the corresponding cusp form, and then constructing a curve from it, is isogenous to the original curve (but not in general isomorphic to it). In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic geometry), for everywhere-regular differential 1-forms. ... In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ...

History

An incorrect version of this theorem was first conjectured by Yutaka Taniyama in September 1955. With Goro Shimura he improved its rigor until 1957. Taniyama committed suicide in 1958; his colleagues guess that his confidence shattered after his repeated failures to solve his conjecture himself. The conjecture was rediscovered by André Weil in 1967, who 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. In the 1970s it became associated with the Langlands program of unifying conjectures in mathematics. Look up theorem in Wiktionary, the free dictionary. ... In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic. ... A sketch of Yutaka Taniyama Yutaka Taniyama (Japanese: è°·å±± 豊 Taniyama Yutaka[1]; November 12, 1927 – November 17, 1958) was a Japanese mathematician, best known for conjecturing, in modern language, automorphic properties of L-functions of elliptic curves over any number field. ... Year 1955 (MCMLV) was a common year starting on Saturday (link displays the 1955 Gregorian calendar). ... Goro Shimura (志村 五郎, 1930 -) is a Japanese-American mathematician, and currently a professor of mathematics at Princeton University. ... Year 1957 (MCMLVII) was a common year starting on Tuesday (link displays the 1957 Gregorian calendar). ... Year 1958 (MCMLVIII) was a common year starting on Wednesday (link will display full calendar) of the Gregorian calendar. ... André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. ... In mathematics, the Langlands program is a web of far-reaching and influential conjectures that connect number theory and the representation theory of certain groups. ...

It attracted considerable interest in the 1980s when Gerhard Frey 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 give rise to a non-modular elliptic curve. Ken Ribet later proved this result. In 1995, Andrew Wiles, with the partial help of Richard Taylor, proved the modularity theorem for semistable elliptic curves, which was strong enough to yield a proof of Fermat's Last Theorem. The 1980s refers to the years from 1980 to 1989. ... Gerhard Frey is a German mathematician, known for his work in number theory. ... Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ... Kenneth Alan Ken Ribet is an American mathematician, currently a professor of mathematics at the University of California, Berkeley. ... Year 1995 (MCMXCV) was a common year starting on Sunday (link will display full 1995 Gregorian calendar). ... For the French mathematician with work in the area of elliptic curves, see André Weil. ... Richard Taylor (born 19 May 1962) is a British mathematician working in the field of number theory. ... In mathematics, a semistable elliptic curve in diophantine geometry is an elliptic curve that has bad reduction only of multiplicative type. ... Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ...

The full modularity theorem was finally proved in 1999 by Breuil, Conrad, Diamond, and Taylor who, building on Wiles' work, incrementally chipped away at the remaining cases until the full result was proved. This article is about the year. ...

Several theorems in number theory 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.) Coprime - Wikipedia /**/ @import /skins-1. ... Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ...

In March 1996 Wiles shared the Wolf Prize with Robert Langlands. Year 1996 (MCMXCVI) was a leap year starting on Monday (link will display full 1996 Gregorian calendar). ... Past winners of the Wolf Prize in Mathematics: 1978 Israel M. Gelfand, Carl L. Siegel 1979 Jean Leray, André Weil 1980 Henri Cartan, Andrei Kolmogorov 1981 Lars Ahlfors, Oscar Zariski 1982 Hassler Whitney, Mark Grigoryevich Krein 1983/4 Shiing S. Chern, Paul Erdős 1984/5 Kunihiko Kodaira, Hans... Robert Langlands (born 1936 in Canada) is one of the most significant mathematicians of the 20th century, with profound insights in number theory and representation theory. ...

References

• Henri Darmon: A Proof of the Full Shimura-Taniyama-Weil Conjecture Is Announced, Notices of the American Mathematical Society, Vol. 46 (1999), No. 11. Contains a gentle introduction to the theorem and an outline of the proof.
• Christophe Breuil, Brian Conrad, Fred Diamond, Richard Taylor: On the modularity of elliptic curves over Q: Wild 3-adic exercises, Journal of the American Mathematical Society 14 (2001), pp. 843–939. Contains the proof of the modularity theorem.
• Barry Mazur, Number theory as gadfly- American Mathematical Monthly, 98 (7), August-September 1991, pp. 593–610, Discusses the Taniyama-Shimura conjecture 3 years before it was proven for infinitely many cases.
• Weil, André Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann. 168 1967 149-156.
• Wiles, Andrew Modular elliptic curves and Fermat's last theorem. Ann. of Math. (2) 141 (1995), no. 3, 443--551.
• Taylor, Richard; Wiles, Andrew Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141 (1995), no. 3, 553--572
• Wiles, Andrew Modular forms, elliptic curves, and Fermat's last theorem. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 243--245, Birkhäuser, Basel, 1995.