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Plane quartics over $$\mathbb {Q}$$ with complex multiplication. (English) Zbl 1409.14051
The 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.

##### 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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