  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 $g \geq 3$. The experiments point to a
stronger notion of convergence than the one provided by the Katz-Sarnak
framework for all curves of genus $\geq 3$. However, for elliptic curves and
for hyperelliptic curves of every genus we prove that this stronger convergence
cannot occur.
