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

Abstract

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

arXiv comment24 pages, 5 figures. We reformulated one of our main results (Theorem 3.1) and added remark 2.6 after one of the referees' questions

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