A classification of 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 $Q$-curve if $E$ is isogenous over $H$ to all its Galois conjugates. We classify $Q$-curves over $H$, relating them with the cohomology group $H^2(H/Q,\pm 1)$. The structures of the abelian varieties over $Q$ obtained from $Q$-curves by restriction of scalars are investigated.