Orbits of rational n-sets of projective spaces under the action of the linear group

Abstract

Let k = F q be a finite field. We enumerate k-rational n-sets of (unordered) points in a projective space P N over k, and we compute the generating function for the numbers of PGL N +1 (k)-orbits of these n-sets. For N = 1, 2 we obtain a formula for these numbers of orbits as a polynomial in q with integer coefficients.

Counting hyperelliptic curves

“…In sections 2, 3 we study when a concrete k-automorphism of P 1 can determine a k-isomorphism between a curve and its hyperelliptic twist (Theorem 3.4). This result provides a way of counting hyp(g) = | Hyp g | by using the techniques of [LMNX02] and [MN07], where a closed formula for the cardinality of the target set PGL 2 (k)\ P 1 2g + 2 (k) was found. As a by-product we obtain also a closed formula for the number of self-dual curves of a given genus (Theorem 5.1).…”

“…By Theorem 2.4 this is the number of k-isomorphy classes of hyperelliptic curves of genus g. In [LMNX02] similar ideas were applied to count the total number of PGL 2 (k)-orbits of rational n-sets of P 1 . In [MN07] a general theory is developed to deal with similar problems in arbitrary dimension. For commodity of the reader we sum up the results we are going to use of these two papers.…”

“…In this paper we find a closed formula for the number hyp(g) of hyperelliptic curves of genus g over a finite field k = F q of odd characteristic (Theorem 4.3). We use a general technique for enumerating PGL 2 (k)-orbits of rational n-sets of P 1 that was developed in [LMNX02] and extended to arbitrary dimension in [MN07]. For n = 2g + 2, each n-set S = {t 1 , .…”

Counting hyperelliptic curves that admit a Koblitz model

“…To obtain our results we use a general technique for enumerating PGL 2 (k)orbits of rational n-sets of P 1 that was developed in [12] and extended to arbitrary dimension in [14]. This technique was used in [15] to obtain a formula for the total number of k-isomorphism classes of hyperelliptic curves.…”