## Journal of the Mathematical Society of Japan

### A classification of $Q$-curves with complex multiplication

Tetsuo NAKAMURA

#### 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

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

Digital Object Identifier
doi:10.2969/jmsj/1191418649

Mathematical Reviews number (MathSciNet)
MR2048478

Zentralblatt MATH identifier
1143.11327

#### 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