## Journal of the Mathematical Society of Japan

- J. Math. Soc. Japan
- Volume 56, Number 2 (2004), 635-648.

### 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.

#### Article information

**Source**

J. Math. Soc. Japan, Volume 56, Number 2 (2004), 635-648.

**Dates**

First available in Project Euclid: 3 October 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.jmsj/1191418649

**Digital Object Identifier**

doi:10.2969/jmsj/1191418649

**Mathematical Reviews number (MathSciNet)**

MR2048478

**Zentralblatt MATH identifier**

1143.11327

**Subjects**

Primary: 11G05: Elliptic curves over global fields [See also 14H52]

Secondary: 11G1O 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22]

**Keywords**

Q-curve elliptic curve complex multiplication embedding problem restriction of scalars

#### Citation

NAKAMURA, Tetsuo. A classification of $Q$ -curves with complex multiplication. J. Math. Soc. Japan 56 (2004), no. 2, 635--648. doi:10.2969/jmsj/1191418649. https://projecteuclid.org/euclid.jmsj/1191418649