Plane quartics over \(\mathbb {Q}\) with complex multiplication.

*(English)*Zbl 1409.14051The paper deals with Jacobians of genus \(3\) curves and sextic CM fields. The authors show that there are exactly \(37\) isomorphism classes of CM fields \(K\) for which there exist principally polarized abelian threefolds \(A/\mathbb{Q}\) with field of moduli \(\mathbb{Q}\) and \(\mathrm{End}(A) \cong \mathcal{O}_{K}\). In the proof of this result they list 37 cyclic sextic CM fields whose maximal orders give rise to CM curves of genus 3 with field of moduli \(\mathbb{Q}\). Some of these curves have been computed before this work, namely hyperelliptic and Picard curves. Thus, this paper completes the list of curves of genus 3 over \(\mathbb{Q}\) whose endomorphism rings over \(\overline{\mathbb{Q}}\) are maximal orders of sextic fields. The authors consider the case of plane quartics with trivial automorphism group. The construction of these curves follows the classical path. They determine first the period matrices, and then they compute corresponding invariants. The curves are reconstructed from rational approximations of these invariants. The resulting equations are given at the end of the paper.

The authors point out that new phenomena might occur for plane quartics. These phenomena do not have an exact equivalent in lower genus, so they would require new theoretical development in order to be fully explained.

The authors point out that new phenomena might occur for plane quartics. These phenomena do not have an exact equivalent in lower genus, so they would require new theoretical development in order to be fully explained.

Reviewer: Tony Ezome (Libreville)

##### MSC:

14H25 | Arithmetic ground fields for curves |

11G15 | Complex multiplication and moduli of abelian varieties |

11Y40 | Algebraic number theory computations |

14H45 | Special algebraic curves and curves of low genus |

14K22 | Complex multiplication and abelian varieties |

14K25 | Theta functions and abelian varieties |

14Q05 | Computational aspects of algebraic curves |

13A50 | Actions of groups on commutative rings; invariant theory |

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\textit{P. Kılıçer} et al., Acta Arith. 185, No. 2, 127--156 (2018; Zbl 1409.14051)

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