An abelian variety of dimension has complex multiplication when its rational endomorphism ring contains a commutative semisimple -algebra of dimension . For an elliptic curve in characteristic zero, this means its geometric endomorphism ring of an elliptic curve is an order in an imaginary quadratic field. The resulting rank-one Tate module representations connect torsion points to class field theory.
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Complex multiplication is a concept from complex number theory that involves multiplying complex numbers. A complex number is expressed in the form \( z = a + bi \), where \( a \) and \( b \) are real numbers, \( i \) is the imaginary unit (defined as \( i^2 = -1 \)), \( a \) is the real part, and \( b \) is the imaginary part.