Editing Modularity theorem (section)

==Statement==
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The [[theorem]] states that any [[elliptic curve]] over <math>\Q</math> can be obtained via a [[rational map]] with [[integer]] [[coefficient]]s from the [[classical modular curve]] {{math|''X''<sub>0</sub>(''N'')}} for some integer {{mvar|N}}; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of level {{mvar|N}}.  If {{mvar|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 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 {{mvar|N}}, a normalized [[newform]] with integer {{mvar|q}}-expansion, followed if need be by an [[Elliptic curve#Isogeny|isogeny]].

===Related statements===
The modularity theorem implies a closely related analytic statement:

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

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

The [[generating function]] of the coefficients {{math|''a''<sub>''n''</sub>}} is then

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

If we make the substitution

:<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'',''τ'')}} of the complex variable {{mvar|τ}}, so the coefficients of the {{mvar|q}}-series are also thought of as the Fourier coefficients of {{mvar|f}}. The function obtained in this way is, remarkably, a [[cusp form]] of weight two and level {{mvar|N}} 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).