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| {{Unreferenced|date=December 2009}}
| | Golda is what's written on my beginning certification although it is not the title on my beginning certificate. To climb is something I really enjoy performing. For years she's been working as a travel agent. For a while I've been in Alaska but I will have to transfer in a year or two.<br><br>Have a look at my blog; clairvoyants [[http://cartoonkorea.com/ce002/1093612 Recommended Web-site]] |
| A '''modular elliptic curve''' is an [[elliptic curve]] ''E'' that admits a parametrisation ''X''<sub>0</sub>(''N'') → ''E'' by a [[modular curve]]. This is not the same as a modular curve that happens to be an elliptic curve, and which could be called an elliptic modular curve. The [[modularity theorem]], also known as the [[Taniyama–Shimura conjecture]], asserts that every elliptic curve defined over the rational numbers is modular.
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| ==Modularity theorem==
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| The [[theorem]] states that any [[elliptic curve]] over '''Q''' can be obtained via a [[rational map]] with [[integer]] [[coefficient]]s from the [[classical modular curve]]
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| :<math>X_0(N)\ </math>
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| 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 [[Elliptic curve#Isogeny|isogeny]].
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| The modularity theorem implies a closely related analytic statement: to an elliptic curve ''E'' over '''Q''' we may attach a corresponding [[L-series of an elliptic curve|L-series]]. The ''L''-series is a [[Dirichlet series]], commonly written
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| :<math>L(s, E) = \sum_{n=1}^\infty \frac{a_n}{n^s}.</math>
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| The [[generating function]] of the coefficients <math>a_n</math> is then
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| :<math>f(q, E) = \sum_{n=1}^\infty a_n q^n.</math> | |
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| If we make the substitution
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| :<math>q = e^{2 \pi i \tau}\ </math>
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| we see that we have written the [[Fourier series|Fourier expansion]] of a function <math>f(\tau, E)</math> of the complex variable ''τ'', so the coefficients of the ''q''-series are also thought of as the Fourier coefficients of <math>f</math>. The function obtained in this way is, remarkably, a [[modular form|cusp form]] of weight two and level ''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.
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| 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).
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| {{DEFAULTSORT:Modular Elliptic Curve}}
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| ==References==
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| *{{Citation | last1=Wiles | first1=Andrew | author1-link=Andrew Wiles | title=Modular elliptic curves and Fermat's last theorem | jstor=2118559 | mr=1333035 | year=1995 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=141 | issue=3 | pages=443–551}}
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| *{{Citation | last1=Wiles | first1=Andrew | author1-link=Andrew Wiles | title=Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) | publisher=Birkhäuser | location=Basel, Boston, Berlin | mr=1403925 | year=1995 | chapter=Modular forms, elliptic curves, and Fermat's last theorem | pages=243–245}}
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| [[Category:Elliptic curves]]
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Golda is what's written on my beginning certification although it is not the title on my beginning certificate. To climb is something I really enjoy performing. For years she's been working as a travel agent. For a while I've been in Alaska but I will have to transfer in a year or two.
Have a look at my blog; clairvoyants [Recommended Web-site]