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Complex Multiplication

Complex multiplication of complex numbers x=a+ib and y=c+id is given by

xy=(a+ib)(c+id)
(1)
=ac+ibc+iad-bd
(2)
=(ac-bd)+i(ad+bc).
(3)

In component form,

(x,y)(x^',y^')=(xx^'-yy^',xy^'+yx^')
(4)

(Krantz 1999, p. 1). The special case of a complex number multiplied by a scalar a is then given by

a(x,y)=(a,0)(x,y)=(ax,ay).
(5)

Surprisingly, complex multiplication can be carried out using only three real multiplications, ac, bd, and (a+b)(c+d) as

Re[(a+ib)(c+id)]=ac-bd
(6)
Im[(a+ib)(c+id)]=(a+b)(c+d)-ac-bd.
(7)

In algebraic geometry, complex multiplication has a distinct meaning. A complex Abelian variety A of dimension g has complex multiplication iff its rational endomorphism algebra End(A) tensor ZQ contains a commutative semisimple Q-subalgebra of dimension 2g (Milne 2020, Proposition 3.3). For a complex elliptic curve, this means that the endomorphism ring is larger than Z.

See also

Complex Addition, Complex Division, Complex Exponentiation, Complex Number, Complex Subtraction, Elliptic Curve, Imaginary Part, Multiplication, Real Part, Siegel Moduli Variety, Sign, Zilber-Pink Conjecture

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References

Cox, D. A. Primes of the Form x2+ny2: Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1997.Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 1, 1999.Milne, J. S. "Complex Multiplication." Version 0.10, 14 Jul 2020. Proposition 3.3. https://www.jmilne.org/math/CourseNotes/CM.pdf.

Referenced on Wolfram|Alpha

Complex Multiplication

Cite this as:

Weisstein, Eric W. "Complex Multiplication." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ComplexMultiplication.html

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