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.

For a given genus $g \geq 1$, we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus g over $\mathbb{F}_q$. As a consequence of Katz–Sarnak theory, we first get for any given $g>0$, any $\varepsilon>0$ and all q large enough, the existence of a curve of genus g over $\mathbb{F}_q$ with at least $1+q+ (2g-\varepsilon) \sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 \sqrt{q}$ valid for $g \geq 3$ and odd $q \geq 11$. Finally, explicit constructions of towers of curves improve this result: We show that the bound $1+q+4 \sqrt{q} -32$ is valid for all $g\ge 2$ and for all q.

We study the problem of efficiently constructing a curve $C$ of genus $2$ over a finite field $\mathbb{F}$ for which either the curve $C$ itself or its Jacobian has a prescribed number $N$ of $\mathbb{F}$-rational points.

In the case of the Jacobian, we show that any ‘CM-construction’ to produce the required genus-$2$ curves necessarily takes time exponential in the size of its input.

On the other hand, we provide an algorithm for producing a genus-$2$ curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applicability of this algorithm by constructing a genus-$2$ curve having exactly $10^{2014}+9703$ (prime) points, and two genus-$2$ curves each having exactly $10^{2013}$ points.

In an appendix we provide a complete parametrization, over an arbitrary base field $k$ of characteristic neither two nor three, of the family of genus-$2$ curves over $k$ that have $k$-rational degree-$3$ maps to elliptic curves, including formulas for the genus-$2$ curves, the associated elliptic curves, and the degree-$3$ maps.

Supplementary materials are available with this article.

three families of pairs of curves are presented; each pair consists of geometrically non-isomorphic curves whose jacobians are isomorphic to one another as unpolarized abelian varieties. each family is parametrized by an open subset of ${\mathbb p}^1$. the first family consists of pairs of genus-2 curves whose equations are given by simple expressions in the parameter; the curves in this family have reducible jacobians. the second family also consists of pairs of genus-2 curves, but generically the curves in this family have absolutely simple jacobians. the third family consists of pairs of genus-3 curves, one member of each pair being a hyperelliptic curve and the other a plane quartic. examples from these families show that in general it is impossible to tell from the jacobian of a genus-2 curve over $\mathbb q$ whether or not the curve has rational points – or indeed whether or not it has real points. the families are constructed using methods that depend on earlier joint work with franck leprévost and bjorn poonen, and on peter bending's explicit description of the curves of genus 2 whose jacobians have real multiplication by $\mathbb z[\sqrt{2}]$.

We prove that if q is a power of an odd prime, then there is no genus-2 curve over $\mathbf{F}_q$ whose Jacobian has characteristic polynomial of Frobenius equal to $x^4 + (2 - 2q)x^2 + q^2$. Our proof uses the Brauer relations in a biquadratic extension of $\mathbb{Q}$ to show that every principally polarized abelian surface over $\mathbf{F}_q$ with the given characteristic polynomial splits over $\mathbf{F}_{q^2}$ as a product of polarized elliptic curves.