Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields

Reynald Lercier (1) , Christophe Ritzenthaler (1) , Florent Rovetta (2) , Jeroen Sijsling (3)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 I2M - Institut de Mathématiques de Marseille
3 Dartmouth College [Hanover]

We study new families of curves that are suitable for efficiently parametrizing their moduli spaces. We explicitly construct such families for smooth plane quartics in order to determine unique representatives for the isomorphism classes of smooth plane quartics over finite fields. In this way, we can visualize the distributions of their traces of Frobenius. This leads to new observations on fluctuations with respect to the limiting symmetry imposed by the theory of Katz and Sarnak.

Mots clés: moduli; families; enumeration; Genus 3 curves; plane quartics; finite fields

Domaines: Géométrie algébrique [math.AG]

Reynald Lercier, Christophe Ritzenthaler, Florent Rovetta, Jeroen Sijsling. Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields. Algorithmic Number Theory Symposium ANTS-XI, Aug 2014, GyeongJu, South Korea, France. pp.128-147, ⟨10.1112/S146115701400031X⟩. ⟨hal-00980244⟩