Products of CM elliptic curves (Q644468)

The present paper concerns for an arbitrary field \(K\) the problem whether a given abelian variety \(A\) over \(K\) is isomorphic to a product of isogenous CM-elliptic curves. Previously ciriteria have been worked out over \(\mathbb C\) by \textit{T. Shioda} and \textit{N. Mitani} in dimension 2 [Lect. Notes Math. 412, 259--287 (1974; Zbl 0302.14011)] and implicitly by \textit{C. Schoen} in higher dimension [J. Reine Angew. Math. 429, 115-123 (1992; Zbl 0744.14030)]. Either approach relies on period lattices. Presently the role of the lattice is replaced by ideal classes in homomorphism rings of elliptic curves. The main advantage of this replacement is that it works over arbitrary fields. As a special case, the author treats the isomorphy problem for two abelian varieties which are already known to be of product type.