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. The theorem states that any elliptic curve over can be obtained via a rational map with integer coefficients from the classical modular curve for some integer ; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of level . If 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 , a normalized newform with integer -expansion, followed if need be by an isogeny. The modularity theorem implies a closely related analytic statement: To each elliptic curve E over we may attach a corresponding L-series. The -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 variable , so the coefficients of the -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 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.