Modularity theorem: Difference between revisions

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==Statement==

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The [[theorem]] states that any [[elliptic curve]] over '''<math>\mathbf{Q'''}</math> can be obtained via a [[rational map]] with [[integer]] [[coefficient]]s from the [[classical modular curve]] <math>X_0(N)</math> for some integer ''<math>N''</math>; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of level ''<math>N''</math>. If ''<math>N''</math> 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 of an elliptic curve|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 [[Elliptic curve#Isogeny|isogeny]].

===Related statements===

The modularity theorem implies a closely related analytic statement:

to an elliptic curve ''E'' over '''<math>\mathbf{Q'''}</math> we may attach a corresponding [[L-series of an elliptic curve|L-series]]. The ''L''-series is a [[Dirichlet series]], commonly written

:<math>L(E, s) = \sum_{n=1}^\infty \frac{a_n}{n^s}.</math>

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:<math>q = e^{2 \pi i \tau}</math>

we see that we have written the [[Fourier series|Fourier expansion]] of a function <math>f(E, \tau)</math> of the complex variable ''τ''<math>\tau</math>, so the coefficients of the ''<math>q''</math>-series are also thought of as the Fourier coefficients of <math>f</math>. The function obtained in this way is, remarkably, a [[cusp form]] of weight two and level ''<math>N''</math> and is also an eigenform (an eigenvector of all [[Hecke operator]]s); this is the '''Hasse–Weil conjecture''', which follows from the modularity theorem.

Some modular forms of weight two, in turn, correspond to [[holomorphic differential]]s 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 [[Elliptic curve#Isogeny|isogenous]] to the original curve (but not, in general, isomorphic to it).