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{{MSC|11G05|11D41,11F11,11F80,11G18}}
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''Shimura–Taniyama–Weil conjecture, Taniyama–Shimura conjecture, Taniyama–Weil conjecture, modularity conjecture''

A conjecture that postulates a deep connection between elliptic curves (cf. [[Elliptic curve|Elliptic curve]]) over the rational numbers and modular forms (cf. [[Modular form|Modular form]]). It has been completely proved thanks to the fundamental work of A. Wiles and R. Taylor
{{Cite|Wi}},
{{Cite|TaWi}}, and its further refinements
{{Cite|Di}},
{{Cite|CoDiTa}}.

Let $\def\G{\Gamma}\G_0(N)$ be the group of matrices in $\def\SL{\textrm{SL}}\SL_2(\Z)$ which are upper-triangular modulo a given positive integer $N$. It acts as a discrete group of Mobius transformations (cf. also [[Discrete group of transformations|Discrete group of transformations]]; [[Fractional-linear mapping|Fractional-linear mapping]]) on the Poincaré upper half-plane $H=\{z\in\C | \textrm{Im}(z) > 0\}$ (cf. also [[Poincaré model|Poincaré model]]). A cusp form of weight $2$ for $\G_0(N)$ is an [[Analytic function|analytic function]] $f$ on $H$ satisfying the relation

$$f\Big(\frac{az+b}{cz+d}\Big) = (cz+d)^2 f(z) \textrm{ for all }
\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\G_0(N),$$
together with suitable growth conditions on the boundary of $H$ (cf. also [[Modular form|Modular form]]). The function $f$ is periodic of period $1$, and it can be written as a [[Fourier series|Fourier series]] in $q=e^{2\pi iz}$ with no constant term: $f(z)=\sum_{n=1}^\infty \def\l{\lambda} \l_n q^n$. The [[Dirichlet series|Dirichlet series]] $L(f,s) = \sum \l_n n^{-s}$ is called the $L$-function attached to $f$ (cf. also [[Fourier coefficients of automorphic forms|Fourier coefficients of automorphic forms]]; [[Dirichlet L-function|Dirichlet $L$-function]]). It is essentially the [[Mellin transform|Mellin transform]] of $f$:

$$\Lambda(f,s):= \G(s)L(f,s) = (2\pi)^2 \int_0^\infty f(iy)y^{s-1}dy.$$
The space of cusp forms of weight $2$ on $\G_0(N)$ is a finite-dimensional [[Vector space|vector space]] and is preserved by the involution $W_N$ defined by $W_N(f)(z) = Nz^2f(-1/(Nz))$. E. Hecke has shown that if $f$ lies in one of the two eigenspaces for this involution (with eigenvalue $w=\pm1$), then $L(f,s)$ satisfies the functional equation $\Lambda(f,s)=-w\Lambda(f,2-s)$, and that $L(f,s)$ has an [[Analytic continuation|analytic continuation]] to all of $\C$.

Let $E$ be an [[Elliptic curve|elliptic curve]] over the rational numbers, and let $L(E,s)$ denote its Hasse–Weil $L$-series. The curve $E$ is said to be modular if there exists a cusp form $f$ of weight $2$ on $\G_0(N)$, for some $N$, such that $L(E,s)=L(f,s)$. The Shimura–Taniyama conjecture asserts that every elliptic curve over $\Q$ is modular. Thus, it gives a framework for proving the analytic continuation and functional equation for $L(E,s)$. It is prototypical of a general relationship between the $L$-functions attached to arithmetic objects and those attached to automorphic forms (cf. also [[Automorphic form|Automorphic form]]), as described in the far-reaching Langlands program.

A. Weil's refinement of the conjecture predicts that the integer $N$ is equal to the arithmetic conductor of $E$ (cf. also [[Elliptic curve|Elliptic curve]]). One now knows that every elliptic curve is modular. Wiles' method proceeds by viewing the Shimura–Taniyama conjecture in a wider framework which predicts the modularity of the (two-dimensional) Galois representations arising from the cohomology of varieties over $\Q$ (cf. also [[Galois theory|Galois theory]]).

The modularity of $E$ can also be formulated as the statement that $E$ is a quotient of the [[Modular curve|modular curve]] $X_0(N)$ over $\Q$; this curve represents the solution to the moduli problem of classifying pairs $(A,C)$ consisting of an elliptic curve $A$ with a distinguished cyclic subgroup $C$ of order $N$. Alternately, if $E$ is modular, then there is a (non-constant) complex-analytic [[Uniformization|uniformization]] $H/\G_0(N) \to E(\C)$.

The importance of the Shimura–Taniyama conjecture is manifold. Firstly, it gives the analytic continuation of $L(E,s)$ for a large class of elliptic curves. The $L$-function itself plays a key role in the study of $E$, most notably through the celebrated Birch–Swinnerton Dyer conjecture. Secondly, the modular curve $X_0(N)$ is endowed with a natural collection of algebraic points arising from the theory of complex multiplication (cf. also [[Elliptic curve|Elliptic curve]]), and the existence of a modular parametrization allows the construction of points on $E$ defined over Abelian extensions of certain imaginary quadratic fields. This fact was exploited by B.H. Gross and D. Zagier and by V.A. Kolyvagin to give strong evidence for the Birch–Swinnerton Dyer conjecture for $E$, under the assumption that $E$ is modular.

The Shimura–Taniyama conjecture admits various generalizations. Replacing $\Q$ by an arbitrary number field $K$, it predicts that an elliptic curve $E$ over $K$ is associated to an automorphic form on $\textrm{GL}_2(K)$. When $K$ is totally real, such an $E$ is often uniformized by a Shimura curve attached to a suitable quaternion algebra (cf. also [[Quaternion|Quaternion]]) over $K$ with exactly one split place at infinity (when $K$ is of odd degree, or when $E$ has at least one prime of multiplicative reduction). In the context of function fields over finite fields, the Shimura–Taniyama conjecture admits an analogue which was established earlier by V.G. Drinfel'd using methods different from those of Wiles.

====References====
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|valign="top"|{{Ref|BrCoDiTa}}||valign="top"|  C. Breuil,   B. Conrad,   F. Diamond,   R. Taylor,   "On the Modularity of elliptic curves" ''J. Amer. Math. Soc.'' 14 (2001), no. 4, 843–939 {{MR|1839918}} {{ZBL|0982.11033}}
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|valign="top"|{{Ref|CoDiTa}}||valign="top"|  B. Conrad,   F. Diamond,   R. Taylor,   "Modularity of certain potentially Barsotti–Tate Galois representations"  ''J. Amer. Math. Soc.''  ''J. Amer. Math. Soc.'' 12 (1999), no. 2, 521–567. {{MR|639612}} {{ZBL|0923.11085}}
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|valign="top"|{{Ref|Di}}||valign="top"|  F. Diamond,   "On deformation rings and Hecke rings"  ''Ann. of Math.'', '''144''' :  1–2  (1996)  pp. 137–166  {{MR|1405946}}  {{ZBL|0867.11032}} 
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|valign="top"|{{Ref|TaWi}}||valign="top"|  R. Taylor,   A. Wiles,   "Ring-theoretic properties of certain Hecke algebras"  ''Ann. of Math.'', '''141''' :  2–3  (1995)  pp. 553–572  {{MR|1333036}}  {{ZBL|0823.11030}} 
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|valign="top"|{{Ref|Wi}}||valign="top"|  A. Wiles,   "Modular elliptic curves and Fermat's last theorem"  ''Ann. of Math.'', '''141''' :  2–3  (1995)  pp. 443–551  {{MR|1333035}}  {{ZBL|0864.11029}} {{ZBL|0823.11029}} 
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