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Examples of CM curves of genus two defined over the reflex field. (English) Zbl 1397.11103
Summary: P. van Wamelen [Math. Comput. 68, No. 225, 307–320 (1999; Zbl 0906.14025)] lists 19 curves of genus two over $$\mathbb{Q}$$ with complex multiplication (CM). However, for each curve, the CM-field turns out to be cyclic Galois over $$\mathbb{Q}$$, 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 K. Lauter and B. Viray [Am. J. Math. 137, No. 2, 497–533 (2015; Zbl 1392.11033)] for Igusa class polynomials.

##### MSC:
 11G15 Complex multiplication and moduli of abelian varieties 11G05 Elliptic curves over global fields 14K22 Complex multiplication and abelian varieties 11G07 Elliptic curves over local fields
##### Keywords:
reflex field; CM curves; Igusa class polynomials
##### Software:
GMP-ECM; RECIP; SageMath; CADO-NFS ; PARI/GP
Full Text:
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