= Complex multiplication
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= CM
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An <abelian variety> of dimension $g$ has complex multiplication when its rational <endomorphism ring> contains a commutative semisimple $\mathbb Q$-algebra of dimension $2g$. 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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