Modularity Theorem - Statement

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

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.

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 variable τ, 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 (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).