Parametrizing the moduli space of curves and applications to smooth   plane quartics over finite fields
  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. 
Reynald Lercier, Christophe Ritzenthaler, Florent Rovetta, and Jeroen   Sijsling
math.AG