Counting hyperelliptic curves

We find a closed formula for the number hyp(g) of hyperelliptic curves of genus g over a finite field k = Fq of odd characteristic. These numbers hyp(g) are expressed as a polynomial in q with integer coefficients that depend on g and the set of divisors of q − 1 and q + 1. As a by-product we obtain a closed formula for the number of self-dual curves of genus g. A hyperelliptic curve is self-dual if it is k-isomorphic to its own hyperelliptic twist. Introduction In this paper we find a closed formula for the number hyp(g) of hyperelliptic curves of genus g over a finite field k = Fq of odd characteristic (Theorem 4.3). We use a general technique for enumerating PGL2(k)-orbits of rational n-sets of P that was developed in [LMNX02] and extended to arbitrary dimension in [MN07]. For n = 2g + 2, each n-set S = {t1, . . . , tn} of P determines a family of hyperelliptic curves of genus g whose Weierstrass points have x-coordinate in S: Cλ,S : y 2 = λ ∏ t∈S, t6=∞ (x− t), λ ∈ k. If S is a rational n-set (i.e. stable under the action of the Galois group Gal(k/k)), the curve Cλ,S is defined over k. These curves fall generically into two different k-isomorphism classes, represented by a curve and its hyperelliptic twist. However, there are n-sets for which these curves are all k-isomorphic; in other words there are self-dual curves that are k-isomorphic to their own hyperelliptic twist. Finally, two different rational n-sets of P determine the same family of hyperelliptic curves up to k-isomorphism if and only if they are in the same orbit under partially supported by grant MTM2006-11391 from the Spanish MEC 1 the natural action of PGL2(k). Summing up, if Hypg is the set of k-isomorphism classes of hyperelliptic curves over k of genus g, and ( P n ) (k) is the set of rational n-sets of P we have a well defined map Hypg −→ PGL2(k)\ ( P 2g + 2 ) (k), sending a curve C to the 2g+2-set of x-coordinates of the Weierstrass points of a Weierstrass model of C. This map is onto and the orbit of each n-set S has either one or two preimages according to Cλ,S being self-dual or not. In sections 2, 3 we study when a concrete k-automorphism of P can determine a k-isomorphism between a curve and its hyperelliptic twist (Theorem 3.4). This result provides a way of counting hyp(g) = |Hypg | by using the techniques of [LMNX02] and [MN07], where a closed formula for the cardinality of the target set PGL2(k)\ ( P 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). Acknowledgement. It is a pleasure to thank Amparo López for providing the key argument to prove Lemma 3.5. Notations. We fix once and for all a finite field k = Fq of odd characteristic p and an algebraic closure k of k. We denote by Gk the Galois group Gal(k/k) and by σ(x) = x the Frobenius automorphism, which is a topological generator of Gk as a profinite group. Also, k2 will denote the unique quadratic extension of k in k. 1 Generalities on hyperelliptic curves Let C be a hyperelliptic curve defined over k; that is, C is a smooth, projective and geometrically irreducible curve defined over k, of genus g ≥ 2, and it admits a degree two morphism π : C −→ P, which is also defined over k. To the non-trivial element of the Galois group of the quadratic extension of function fields, k(C)/π(k(P)), it corresponds an involution, ι : C −→ C, which is called the hyperelliptic involution of C. Let us recall some basic properties of ι. Theorem 1.1. Let π1, π2 : C −→ P be two k-morphisms of degree two. Then, there exists a unique k-automorphism γ of P such that π2 = γ ◦π1. In particular, the involution ι is canonical, its fixed points are the Weierstrass points of C, and they are the ramification points of any morphism of degree two from C to P.