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| {{about|a topic in the theory of [[elliptic curves]]|information about multiplication of complex numbers|complex numbers}}
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| In [[mathematics]], '''complex multiplication''' is the theory of [[elliptic curve]]s ''E'' that have an [[endomorphism ring]] larger than the [[integer]]s; and also the theory in higher dimensions of [[abelian variety|abelian varieties]] ''A'' having ''enough'' endomorphisms in a certain precise sense (it roughly means that the action on the [[tangent space]] at the [[identity element]] of ''A'' is a [[direct sum of modules|direct sum]] of one-dimensional [[module (mathematics)|modules]]). Put another way, it contains the theory of [[elliptic function]]s with extra symmetries, such as are visible when the [[period lattice]] is the [[Gaussian integer]] [[Lattice (group)|lattice]] or [[Eisenstein integer]] lattice.
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| It has an aspect belonging to the theory of [[special function]]s, because such elliptic functions, or [[abelian function]]s of [[several complex variables]], are then 'very special' functions satisfying extra identities and taking explicitly calculable special values at particular points. It has also turned out to be a central theme in [[algebraic number theory]], allowing some features of the theory of [[cyclotomic field]]s to be carried over to wider areas of application.
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| [[David Hilbert]] is said to have remarked that the theory of complex multiplication of elliptic curves was not only the most beautiful part of mathematics but of all science.<ref>{{Citation
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| | last=Reid
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| | first=Constance
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| | author-link=Constance Reid
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| | title=Hilbert
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| | publisher=Springer
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| | isbn=978-0-387-94674-0
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| | year=1996
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| | page=200}}</ref>
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| ==Example==
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| An example of an elliptic curve with complex multiplication is
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| :'''C'''/'''Z'''[''i'']θ
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| where '''Z'''[''i''] is the [[Gaussian integer]] ring, and θ is any non-zero complex number. Any such complex [[torus]] has the Gaussian integers as endomorphism ring. It is known that the corresponding curves can all be written as
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| : ''Y''<sup>2</sup> = 4''X''<sup>3</sup> − ''aX'',
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| having an order 4 [[automorphism]] sending
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| : ''Y'' → −''iY'', ''X'' → −''X''
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| in line with the action of ''i'' on the [[Weierstrass elliptic function]]s.
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| This is a typical example of an elliptic curve with complex multiplication. Over the complex numbers, all elliptic curves with complex multiplication can be similarly constructed. That is, as quotients of the complex plane by some [[order (ring theory)|order]] in the [[ring of integers]] in an imaginary [[quadratic field]].
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| ==Abstract theory of endomorphisms==
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| The ring of endomorphisms of an elliptic curve can be of one of three forms:the integers '''Z'''; an [[Order (ring theory)|order]] in an [[imaginary quadratic number field]]; or an order in a definite [[quaternion algebra]] over '''Q'''.<ref>Silverman (1989) p.102</ref>
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| When the field of definition is a [[finite field]], there are always non-trivial endomorphisms of an elliptic curve, coming from the [[Frobenius map]], so the ''complex multiplication'' case is in a sense typical (and the terminology isn't often applied). But when the base field is a number field, complex multiplication is the exception. It is known that, in a general sense, the case of complex multiplication is the hardest to resolve for the [[Hodge conjecture]].
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| ==Kronecker and abelian extensions==
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| [[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]].
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| 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''.<ref name=S295>Serre (1967) p.295</ref>
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| 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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| ==Sample consequence==
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| It is no accident that
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| : <math>e^{\pi \sqrt{163}} = 262537412640768743.99999999999925007\dots\,</math>
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| or equivalently,
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| : <math>e^{\pi \sqrt{163}} = 640320^3+743.99999999999925007\dots\,</math>
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| is so close to an integer. This remarkable fact is explained by the theory of complex multiplication, together with some knowledge of [[modular forms]], and the fact that
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| : <math>\mathbf{Z}\left[ \frac{1+\sqrt{-163}}{2}\right]</math>
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| is a [[unique factorization domain]].
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| Here <math>(1+\sqrt{-163})/2</math> satisfies α² = α − 41. In general, ''S''[α] denotes the set of all [[polynomial]] expressions in α with coefficients in ''S'', which is the smallest ring containing α and ''S''. Because α satisfies this quadratic equation, the required polynomials can be limited to degree one.
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| Alternatively,
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| : <math>e^{\pi \sqrt{163}} = 12^3(231^2-1)^3+743.99999999999925007\dots\,</math>
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| an internal structure due to certain Eisenstein series, and with similar simple expressions for the other [[Heegner number]]s.
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| ==Singular moduli==
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| The points of the upper half-plane τ which correspond to the period ratios of elliptic curves over the complex numbers with complex multiplication are precisely the imaginary quadratic numbers.<ref>Silverman (1986) p.339</ref> The corresponding [[Elliptic modular function|modular invariant]]s ''j''(τ) are the '''singular moduli''', coming from an older terminology in which "singular" referred to the property of having non-trivial endomorphisms rather than referring to a [[singular curve]].<ref>Silverman (1994) p.104</ref>
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| The [[modular function]] ''j''(τ) is algebraic on imaginary quadratic numbers τ:<ref name=S293>Serre (1967) p.293</ref> these are the only algebraic numbers in the upper half-plane for which ''j'' is algebraic.<ref>{{cite book | first=Alan | last=Baker | authorlink=Alan Baker (mathematician) | title=Transcendental Number Theory | publisher=[[Cambridge University Press]] | year=1975 | isbn=0-521-20461-5 | zbl=0297.10013 | page=56 }}</ref>
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| If Λ is a lattice with period ratio τ then we write ''j''(Λ) for ''j''(τ). If further Λ is an ideal '''a''' in a ring of integers ''O'' of a quadratic imaginary field ''K'' then we write ''j''('''a''') for the corresponding singular modulus. The values ''j''('''a''') are then real algebraic integers, and generate the [[Hilbert class field]] ''H'' of ''K'': the [[field extension]] degree [''H'':''K''] = ''h'' is the class number of ''K'' and the ''H''/''K'' is a [[Galois extension]] with [[Galois group]] isomorphic to the [[ideal class group]] of ''K''. The class group acts on the values ''j''('''a''') by ['''b'''] : ''j''('''a''') → ''j''('''ab''').
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| In particular, if ''K'' has class number one, then ''j''('''a''') = ''j''(''O'') is a rational integer: for example, ''j''('''Z'''[i]) = ''j''(i) = 1728.
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| ==See also==
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| *[[Abelian variety of CM-type]], higher dimensions
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| *[[Algebraic Hecke character]]
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| *[[Heegner point]]
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| *[[Hilbert's twelfth problem]]
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| *[[Lubin–Tate formal group]], [[local field]]s
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| *[[Drinfeld shtuka]], [[global function field]] case
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * Borel, A.; Chowla, S.; Herz, C. S.; Iwasawa, K.; Serre, J.-P. ''Seminar on complex multiplication''. Seminar held at the Institute for Advanced Study, Princeton, N.J., 1957-58. Lecture Notes in Mathematics, No. 21 Springer-Verlag, Berlin-New York, 1966
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| * {{cite book | last=Husemöller | first=Dale H. | title=Elliptic curves | others=With an appendix by Ruth Lawrence | series=Graduate Texts in Mathematics | volume=111 |publisher=[[Springer-Verlag]] | year=1987 | isbn=0-387-96371-5 | zbl=0605.14032 }}
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| * {{cite book | last=Lang | first=Serge | authorlink=Serge Lang | title=Complex multiplication | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | volume=255 | publisher=[[Springer-Verlag]] | location=New York | year=1983 | isbn=0-387-90786-6 | zbl=0536.14029 }}
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| * {{cite book | last=Serre | first=J.-P. | authorlink=Jean-Pierre Serre | chapter=XIII. Complex multiplication | pages=292–296 | editor1-first=J.W.S. | editor1-last=Cassels | editor1-link=J. W. S. Cassels | editor2-first=Albrecht | editor2-last=Fröhlich | editor2-link=Albrecht Fröhlich | title=Algebraic Number Theory | year=1967 | publisher=Academic Press | zbl= }}
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| *{{cite book | last=Shimura | first=Goro | authorlink=Goro Shimura | title=Introduction to the arithmetic theory of automorphic functions | publisher=Iwanami Shoten | location=Tokyo | series=Publications of the Mathematical Society of Japan | year=1971 | volume=11 | zbl=0221.10029 }}
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| * {{cite book | last=Shimura | first=Goro | authorlink=Goro Shimura | title=Abelian varieties with complex multiplication and modular functions | series=Princeton Mathematical Series | volume=46 | publisher=[[Princeton University Press]] | location=Princeton, NJ | year=1998 | isbn=0-691-01656-9 | zbl=0908.11023 }}
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| * {{cite book | first=Joseph H. | last=Silverman | authorlink=Joseph H. Silverman | title=The Arithmetic of Elliptic Curves | series=[[Graduate Texts in Mathematics]] | publisher=[[Springer-Verlag]] | volume=106 | year=1986 | isbn=0-387-96203-4 | zbl=0585.14026 }}
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| * {{cite book | first=Joseph H. | last=Silverman | authorlink=Joseph H. Silverman | title=Advanced Topics in the Arithmetic of Elliptic Curves | series=[[Graduate Texts in Mathematics]] | publisher=[[Springer-Verlag]] | volume=151 | year=1994 | isbn=0-387-94328-5 | zbl=0911.14015 }}
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| ==External links==
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| * [http://planetmath.org/encyclopedia/ComplexMultiplication.html Complex multiplication] from [[PlanetMath|PlanetMath.org]]
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| * [http://planetmath.org/encyclopedia/ExamplesOfEllipticCurvesWithComplexMultiplication.html Examples of elliptic curves with complex multiplication] from [[PlanetMath|PlanetMath.org]]
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| * {{cite journal | title = Galois Representations and Modular Forms | id = {{citeseerx|10.1.1.125.6114}} | authorlink = Kenneth Alan Ribet | first = Kenneth A. | last = Ribet | journal = [[Bulletin of the American Mathematical Society]] | volume = 32 | issue = 4 | month = October | year = 1995 | pages = 375–402 }}
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| {{DEFAULTSORT:Complex Multiplication}}
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| [[Category:Abelian varieties]]
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| [[Category:Elliptic functions]]
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| [[Category:Class field theory]]
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