Refinements of Katz-Sarnak theory for the number of points on curves over finite fields

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.