The theorem states that any elliptic curve over Q can be obtained via a rational map with integer coefficients from the classical modular curve 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, 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. 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 is then If we make the substitution we see that we have written the Fourier expansion of a function of the complex variableτ, so the coefficients of the q-series are also thought of as the Fourier coefficients of. The function obtained in this way is, remarkably, a cusp form of weight two and level N and is also an eigenform ; 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 be written as a product of irreducible Abelian varieties, corresponding to Hecke eigenforms of weight 2. The 1-dimensional factors are elliptic curves. The curve obtained by finding the corresponding cusp form, and then constructing a curve from it, is isogenous to the original curve. stated a preliminary 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. rediscovered the conjecture, and showed that it would follow from the 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 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 identified a missing link in Frey's original work, followed two years later by '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' ex-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 was completely inaccessible". , 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,, and who, building on Wiles' 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 coprimen-th powers, n ≥ 3.
