Currently available as a preprint and as an official version.

This paper goes beyond Katz–Sarnak theory on the distribution of curves over finite fields according to their number of rational points, theoretically, experimentally and conjecturally. In particular, we give a formula for the limits of the moments measuring the asymmetry of this distribution for (non-hyperelliptic) curves of genus g ≥ 3. The experiments point to a stronger notion of convergence than the one provided by the Katz–Sarnak framework for all curves of genus g ≥ 3. However, for elliptic curves and for hyperelliptic curves of every genus we prove that this stronger convergence cannot occur.