Editing Complex multiplication

{{Short description|Theory of a class of elliptic curves}}
{{about|a topic in the theory of [[elliptic curves]]|information about multiplication of complex numbers|complex numbers}}
In [[mathematics]], '''complex multiplication''' ('''CM''') is the theory of [[elliptic curve]]s ''E'' that have an [[endomorphism ring]] larger than the [[integer]]s.{{sfn|Silverman|2009|p=69|loc=Remark 4.3}} 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.

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. [[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
| last=Reid
| first=Constance
| author-link=Constance Reid
| title=Hilbert
| publisher=Springer
| isbn=978-0-387-94674-0
| year=1996
| page=[https://archive.org/details/hilbert0000reid/page/200 200]
| url=https://archive.org/details/hilbert0000reid/page/200
}}</ref>

There is also the [[Complex multiplication of abelian varieties|higher-dimensional complex multiplication theory]] of [[abelian variety|abelian varieties]] ''A'' having ''enough'' endomorphisms in a certain precise sense, roughly 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]].

==Example of the imaginary quadratic field extension==

[[Image:Lattice torsion points.svg|right|thumb|300px|An elliptic curve over the complex numbers is obtained as a quotient of the complex plane by a lattice Λ, here spanned by two fundamental periods ''ω''<sub>1</sub> and ''ω''<sub>2</sub>. The four-torsion is also shown, corresponding to the lattice 1/4 Λ containing Λ.

The example of an elliptic curve corresponding to the Gaussian integers occurs when ω<sub>2</sub> = ''i'' ''ω''<sub>1</sub>.]]

Consider an imaginary quadratic field <math display="inline">K = \Q\left(\sqrt{-d}\right) , \, d \in \Z, d > 0</math>.
An elliptic function <math>f</math> is said to have '''complex multiplication''' if there is an algebraic relation between <math>f(z)</math> and <math>f(\lambda z)</math> for all <math>\lambda</math> in <math>K</math>.

Conversely, Kronecker conjectured – in what became known as the ''[[Kronecker Jugendtraum]]'' – that every abelian extension of <math>K</math> '''could be obtained''' by the (roots of the) equation of a suitable elliptic curve with complex multiplication.  To this day this remains one of the few cases of [[Hilbert's twelfth problem]] which has actually been solved.

An example of an elliptic curve with complex multiplication is

:<math>\mathbb{C}/ (\theta \mathbb{Z}[i])</math>

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

:<math>Y^2 = 4X^3 - aX</math>

for some <math>a \in \mathbb{C} </math>, which demonstrably has two conjugate order-4 [[automorphism]]s sending

:<math>Y \to \pm iY,\quad X \to -X</math>

in line with the action of ''i'' on the [[Weierstrass elliptic function]]s.

More generally, consider the lattice Λ, an additive group in the complex plane, generated by <math>\omega_1,\omega_2</math>. Then we define the Weierstrass function of the variable <math>z</math> in <math>\mathbb{C}</math> as follows:
:<math>\wp(z;\Lambda) = \wp(z;\omega_1,\omega_2) = \frac{1}{z^2} + \sum_{(m,n)\ne (0,0)} \left\{\frac{1}{(z+m\omega_1+n\omega_2)^2} - \frac{1}{\left(m\omega_1+n\omega_2\right)^2}\right\},</math>
and
:<math>g_2 = 60\sum_{(m,n) \neq (0,0)} (m\omega_1+n\omega_2)^{-4}</math>
:<math>g_3 =140\sum_{(m,n) \neq (0,0)} (m\omega_1+n\omega_2)^{-6}.</math>
Let <math>\wp'</math> be the derivative of <math>\wp</math>. Then we obtain an isomorphism of complex Lie groups:
:<math>w\mapsto(\wp(w):\wp'(w):1) \in \mathbb{P}^2(\mathbb{C}) </math>
from the complex torus group <math>\mathbb{C}/\Lambda</math> to the projective elliptic curve defined in homogeneous coordinates by
:<math>E = \left\{ (x:y:z) \in \mathbb{C}^3 \mid y^2z = 4x^3 - g_2xz^2 - g_3 z^3 \right\} </math>
and where the point at infinity, the zero element of the group law of the elliptic curve, is by convention taken to be <math>(0:1:0)</math>.  
If the lattice defining the elliptic curve is actually preserved under multiplication by (possibly a proper subring of) the ring of integers <math>\mathfrak{o}_K</math> of <math>K</math>, then the ring of analytic automorphisms of <math> E = \mathbb{C}/\Lambda</math> turns out to be isomorphic to this (sub)ring. 

If we rewrite <math>\tau = \omega_1/\omega_2</math> where <math>\operatorname{Im}\tau > 0</math> and <math>\Delta(\Lambda) = g_2(\Lambda)^3 - 27g_3(\Lambda)^2</math>, then
:<math> j(\tau)=j(E)=j(\Lambda)=2^63^3g_2(\Lambda)^3/\Delta(\Lambda)\ .</math>
This means that the [[j-invariant]] of <math>E</math> is an [[algebraic number]] – lying in <math>K</math> – if <math>E</math> has complex multiplication.

==Abstract theory of endomorphisms==
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'''.{{sfn|Silverman|1986|p=102}}

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 every such curve has ''complex multiplication'' (and the terminology is not 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]].

==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.

==Sample consequence==

It is no accident that [[Ramanujan's constant]], the [[transcendental number]]<ref>{{MathWorld|title=Transcendental Number|urlname=TranscendentalNumber}} gives <math>e^{\pi\sqrt{d}}, d \in Z^*</math>, based on
Nesterenko, Yu. V. "On Algebraic Independence of the Components of Solutions of a System of Linear Differential Equations." Izv. Akad. Nauk SSSR, Ser. Mat. 38, 495–512, 1974. English translation in Math. USSR 8, 501–518, 1974.</ref>

: <math>e^{\pi \sqrt{163}} = 262537412640768743.99999999999925007\dots\,</math>
or equivalently,

: <math>e^{\pi \sqrt{163}} = 640320^3+743.99999999999925007\dots\,</math>

is an [[almost integer]], in that it is [[Mathematical coincidence#Containing pi or e and number 163|very close]] to an [[integer]].<ref>[http://mathworld.wolfram.com/RamanujanConstant.html Ramanujan Constant – from Wolfram MathWorld<!-- Bot-generated title -->]</ref> This remarkable fact is explained by the theory of complex multiplication, together with some knowledge of [[modular forms]], and the fact that
: <math>\mathbf{Z}\left[ \frac{1+\sqrt{-163}}{2}\right]</math>
is a [[unique factorization domain]].

Here <math>(1+\sqrt{-163})/2</math> satisfies {{nowrap|1=''α''<sup>2</sup> = ''α'' &minus; 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.

Alternatively,

: <math>e^{\pi \sqrt{163}} = 12^3(231^2-1)^3+743.99999999999925007\dots\,</math>

an internal structure due to certain [[Eisenstein series]], and with similar simple expressions for the other [[Heegner number]]s.

==Singular moduli==
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.{{sfn|Silverman|1986|p=339}} 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]].{{sfn|Silverman|1994|p=104}}

The [[modular function]] ''j''(''τ'') is algebraic on imaginary quadratic numbers ''τ'':{{sfn|Serre|1967|p=293}} these are the only algebraic numbers in the upper half-plane for which ''j'' is algebraic.<ref>{{cite book | first=Alan | last=Baker | author-link=Alan Baker (mathematician) | title=Transcendental Number Theory | publisher=[[Cambridge University Press]] | year=1975 | isbn=0-521-20461-5 | zbl=0297.10013 | page=56 }}</ref>

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 the ring of integers ''O<sub>K</sub>'' 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''').

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.

==See also==
* [[Algebraic Hecke character]]
* [[Heegner point]]
* [[Hilbert's twelfth problem]]
* [[Lubin–Tate formal group]], [[local field]]s
* [[Drinfeld shtuka]], [[global function field]] case
* [[Wiles's proof of Fermat's Last Theorem]]

==Citations==
{{reflist}}

==References==
{{refbegin|2}}
* 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
* {{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 }}
* {{cite book | last=Lang | first=Serge | author-link=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 | url-access=registration | url=https://archive.org/details/complexmultiplic0000lang }}
* {{cite book | last=Serre | first=J.-P. | author-link=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 }}
*{{cite book | last=Shimura | first=Goro  | author-link=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 }}
* {{cite book | last=Shimura | first=Goro  | author-link=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 }}
* {{cite book | first=Joseph H. | last=Silverman | author-link=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  }}
* {{cite book | last=Silverman | first=Joseph H. | author-link=Joseph H. Silverman| title = The Arithmetic of Elliptic Curves | series=Graduate Texts in Mathematics | year=2009 | volume=106 | edition=2nd | publisher=[[Springer Science+Business Media|Springer Science]] | doi=10.1007/978-0-387-09494-6 | url=https://link.springer.com/book/10.1007/978-0-387-09494-6 | isbn=978-0-387-09493-9}}
* {{cite book | first=Joseph H. | last=Silverman | author-link=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 }}
{{refend}}

==External links==
* [http://planetmath.org/ComplexMultiplication Complex multiplication] from [[PlanetMath|PlanetMath.org]]
* [http://planetmath.org/ExamplesOfEllipticCurvesWithComplexMultiplication Examples of elliptic curves with complex multiplication] from [[PlanetMath|PlanetMath.org]]
* {{cite journal | title = Galois Representations and Modular Forms | citeseerx = 10.1.1.125.6114 | author-link = Kenneth Alan Ribet | first = Kenneth A. | last = Ribet | journal = [[Bulletin of the American Mathematical Society]] | volume = 32 | issue = 4 |date=October 1995 | pages = 375–402 | doi=10.1090/s0273-0979-1995-00616-6| arxiv = math/9503219 | s2cid = 16786407 }}

{{DEFAULTSORT:Complex Multiplication}}
[[Category:Abelian varieties]]
[[Category:Elliptic functions]]
[[Category:Class field theory]]