A classification of \bm{Q}-curves with complex multiplication

Abstract

Let H be the Hilbert class field of an imaginary quadratic field K. An elliptic curve E over H with complex multiplication by K is called a \bm{Q}-curve if E is isogenous over H to all its Galois conjugates. We classify \bm{Q}-curves over H, relating them with the cohomology group H²(H/\bm{Q}, ± 1). The structures of the abelian varieties over \bm{Q} obtained from \bm{Q}-curves by restriction of scalars are investigated.