Journal of the Mathematical Society of Japan

A classification of Q-curves with complex multiplication

Tetsuo NAKAMURA

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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 H2(H/Q,±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


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