Modularity theorem

The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem. Later, a series of papers by Wiles's former students Brian Conrad, Fred Diamond and Richard Taylor, culminating in a joint paper with Christophe Breuil, extended Wiles's techniques to prove the full modularity theorem in 2001.
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 formula_1 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.
Related statements.
The modularity theorem implies a closely related analytic statement:
to an elliptic curve "E" over Q we may attach a corresponding L-series. The "L"-series is a Dirichlet series, commonly written
The generating function of the coefficients formula_3 is then
If we make the substitution
we see that we have written the Fourier expansion of a function formula_6 of the complex variable "τ", so the coefficients of the "q"-series are also thought of as the Fourier coefficients of formula_7. The function obtained in this way is, remarkably, 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.
Some modular forms of weight 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 obtained 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).
History.
Yutaka Taniyama stated a preliminary (slightly incorrect) version of the conjecture at the 1955 international symposium on algebraic number theory in Tokyo and Nikkō. Goro Shimura and Taniyama worked on improving its rigor until 1957. André Weil 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 Taniyama–Shimura–Weil conjecture became a part of the Langlands program.
The conjecture attracted considerable interest when Gerhard Frey suggested that it 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 Jean-Pierre Serre identified a missing link (now known as the epsilon conjecture or Ribet's theorem) in Frey's original work, followed two years later by Ken Ribet's completion of a proof of the epsilon conjecture.
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. For example, Wiles's Ph.D. supervisor 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, with some help from Richard Taylor, proved the Taniyama–Shimura–Weil conjecture for all semistable elliptic curves, which he used to prove Fermat's Last Theorem, and the full Taniyama–Shimura–Weil conjecture was finally proved by Diamond, Conrad, Diamond & Taylor, and Breuil, Conrad, Diamond & Taylor who, building on Wiles's work, incrementally chipped away at the remaining cases until the full result was proved. 
Once fully proven, the conjecture became known as the modularity theorem.
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.)
Generalizations.
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. However, Freitas, Le Hung & Siksek proved that elliptic curves defined over real quadratic fields are modular.