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In mathematics, the modularity theorem (formerly called the Taniyama–Shimura–Weil conjecture and several related names) 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, Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor extended Wiles' techniques to prove the full modularity theorem in 2001. The modularity theorem is a special case of more general conjectures due to Robert Langlands. The Langlands program seeks to attach an automorphic form or automorphic representation (a suitable generalization of a modular form) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a number field. Most cases of these extended conjectures have not yet been proved. However, proved that elliptic curves defined over real quadratic fields are modular. ==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). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Modularity theorem」の詳細全文を読む スポンサード リンク
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