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Complex multiplication
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==Kronecker and abelian extensions== {{further|Hilbert's twelfth problem}} [[Leopold Kronecker|Kronecker]] first postulated that the values of [[elliptic function]]s at torsion points should be enough to generate all [[abelian extension]]s for imaginary quadratic fields, an idea that went back to [[Gotthold Eisenstein|Eisenstein]] in some cases, and even to [[Carl Friedrich Gauss|Gauss]]. This became known as the ''[[Kronecker Jugendtraum]]''; and was certainly what had prompted Hilbert's remark above, since it makes explicit [[class field theory]] in the way the [[roots of unity]] do for abelian extensions of the [[rational number|rational number field]], via [[Shimura's reciprocity law]]. Indeed, let ''K'' be an imaginary quadratic field with class field ''H''. Let ''E'' be an elliptic curve with complex multiplication by the integers of ''K'', defined over ''H''. Then the [[maximal abelian extension]] of ''K'' is generated by the ''x''-coordinates of the points of finite order on some Weierstrass model for ''E'' over ''H''.{{sfn|Serre|1967|p=295}} Many generalisations have been sought of Kronecker's ideas; they do however lie somewhat obliquely to the main thrust of the [[Langlands philosophy]], and there is no definitive statement currently known.
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