A classification of Q-curves with complex multiplication

摘要

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, +- 1). The structures of the abelian varieties over Q obtained from Q-curves by restriction of scalars are investigated.