23.1 Isogenies between Elliptic Curves with Complex Multiplication

Let E/k be an elliptic curve with CM by an order O of discriminant D in an imaginary quadratic field K, and let ` be a prime not equal to the characteristic of k. The roots of Φ`(j(E), Y ) correspond to ` + 1 distinct `-isogenies from E to elliptic curves E′, not all of which may be defined over k; this depends on whether the roots lie in k or a proper extension of k. Of these ` + 1 roots, 0, 1, or 2 may correspond to elliptic curves that also have CM by O, depending on whether ` divides the conductor [OK : O] and whether ` is inert, ramified, or split in K, as you proved on Problem Set 10. We note that any such curves can be defined over k, since the set EllO(k) is either empty or includes the j-invariant of every elliptic curve with CM by O. This is clear in characteristic 0, since the ring class field for O is the splitting field of HD(X) over Q (and over K). For finite fields Fp it follows from Theorem 22.10, and in fact it holds for all fields of characteristic p. But what about elliptic curves that that are `-isogenous to E but don’t have CM by O? We know that over a suitable extension of k at least `− 1 such curves exist. These elliptic curves have CM by a different imaginary quadratic order O′ in the same field K, and O′ either contains or is contained by O, with index `.