Plane quartics over $\mathbb {Q}$ with complex multiplication

Pinar K{\i}l\i\c{c}er (1) , Hugo Labrande (2) , Reynald Lercier (3) , Christophe Ritzenthaler (3, 4) , Jeroen Sijsling (3) , Marco Streng (5)

1 Institut für Mathematik [Oldenburg]
2 University of Calgary
3 IRMAR - Institut de Recherche Mathématique de Rennes
4 I2M - Institut de Mathématiques de Marseille
5 Universiteit Leiden = Leiden University

We give examples of smooth plane quartics over QQQ with complex multiplication over Q¯¯¯¯Q¯\overline{Q} by a maximal order with primitive CM type. We describe the required algorithms as we go, these involve the reduction of period matrices, the fast computation of Dixmier-Ohno invariants, and reconstruction from these invariants. Finally, we discuss some of the reduction properties of the curves that we obtain.

Mots clés: Plane quartics; Explicit aspects; Genus 3; Complex multiplication

Domaines: Géométrie algébrique [math.AG]; Théorie des nombres [math.NT]

Pinar K{\i}l\i\c{c}er, Hugo Labrande, Reynald Lercier, Christophe Ritzenthaler, Jeroen Sijsling, et al.. Plane quartics over $\mathbb {Q}$ with complex multiplication. Acta Arithmetica, 2018, 185 (2), pp.127-156. ⟨10.4064/aa170227-16-3⟩. ⟨hal-01856433⟩