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Degree of isogenies of elliptic curves with complex multiplication. (English) Zbl 0943.11031

Let \(E\) be an elliptic curve with complex multiplication such that \(\operatorname {End} E\cong O_f\), where \(O_f= \mathbb{Z}+ fO_K\) and \(K= \mathbb{Q} (\sqrt{D})\). Let \(F:= \mathbb{Q}(j(E))\) and suppose that \(E\) is defined over \(F\). The author gives a complete description of those elliptic curves \(E'\) that are \(F\)-isogenous to \(E\) and calculates all possible degrees \(N\) of cyclic isogenies \(E'\to E\) defined over \(F\). The degree \(N\) of a cyclic isogeny \(E\to E'\) is (roughly speaking) the number of divisors of \(f^2 d_K\) where \(d_K\) denotes the discriminant of \(K\).

MSC:

11G05 Elliptic curves over global fields
11G15 Complex multiplication and moduli of abelian varieties
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