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Examples of CM curves of genus two defined over the reflex field

Part of: Arithmetic algebraic geometry Computational number theory Abelian varieties and schemes

Published online by Cambridge University Press:  01 August 2015

Florian Bouyer  and
Marco Streng
Florian Bouyer
Affiliation:
University of Warwick, United Kingdom email F.Bouyer@Warwick.ac.uk
Marco Streng
Affiliation:
Universiteit Leiden, Netherlands email Marco.Streng@gmail.com

Abstract

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Van Wamelen [Math. Comp. 68 (1999) no. 225, 307–320] lists 19 curves of genus two over $\mathbf{Q}$ with complex multiplication (CM). However, for each curve, the CM-field turns out to be cyclic Galois over $\mathbf{Q}$, and the generic case of a non-Galois quartic CM-field did not feature in this list. The reason is that the field of definition in that case always contains the real quadratic subfield of the reflex field.

We extend Van Wamelen’s list to include curves of genus two defined over this real quadratic field. Our list therefore contains the smallest ‘generic’ examples of CM curves of genus two.

We explain our methods for obtaining this list, including a new height-reduction algorithm for arbitrary hyperelliptic curves over totally real number fields. Unlike Van Wamelen, we also give a proof of our list, which is made possible by our implementation of denominator bounds of Lauter and Viray for Igusa class polynomials.

MSC classification

Primary: 11G15: Complex multiplication and moduli of abelian varieties 14K22: Complex multiplication
Secondary: 11Y99: None of the above, but in this section 11G05: Elliptic curves over global fields 11G07: Elliptic curves over local fields
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015 , pp. 507 - 538
Copyright
© The Author(s) 2015 

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