Curves of genus two over fields of even characteristic

Abstract

Abstract. In this paper we classify curves of genus two over a perfect field k of characteristic two. We find rational models of curves with a given arithmetic structure for the ramification divisor and we give necessary and sufficient conditions for two models of the same type to be k-isomorphic. As a consequence, we obtain an explicit formula for the number of k-isomorphism classes of curves of genus two over a finite field. Moreover, we prove that the field of moduli of any curve coincides with its field of definition, by exhibiting rational models of curves with any prescribed value of their Igusa invariants. Finally, we use cohomological methods to find, for each rational model, an explicit description of its twists. In this way, we obtain a parameterization of all k-isomorphism classes of curves of genus two in terms of geometric and arithmetic invariants.

Isomorphism classes of hyperelliptic curves of genus 2 over finite fields with characteristic 2

“…[13], Cardona et al consider the general curves of genus two, so our results are different from theirs, our results are 2q 3 +q 2 −q+· · ·( see the following Theorem 4.6), and their results are 2q 3 +q 2 +q+· · ·(see Theorem 17 in ref. [13]). The same things happened for odd characteristics(see refs.…”

“…For char(F q ) = 2, Choie and Yun [10] gave some upper and lower bounds for the number of isomorphism classes of hyperelliptic curves of genus 2, and Choie and Jeong [11] obtained some further partial results about it, but the most difficult case remains open. For the number of isomorphism classes of general curves of genus 2, Cardona [12] and Cardona et al [13] also showed the formulae for odd characteristic and even characteristic separately.…”

A Review on the Isomorphism Classes of Hyperelliptic Curves of Genus 2 over Finite Fields Admitting a Weierstrass Point

“…These results are obtained in [17], [18] and [22]. In [14] and [15] the number of isomorphism classes of hyperelliptic curves of genus 2 without assuming the existence of a Weierstrass point for a finite field of arbitrary characteristic is given. Let H,H ∈ W be two curves given by the corresponding equations of the form (1).…”

Families of genus 2 curves with small embedding degree

“…In Theorem 1, we use the results of [17,14,6] for curves whose non-ordinary Jacobian has 2-rank 1, letting C be a curve of genus 2 over F q of the form y 2 + xy = ax 5 + bx 3 + cx 2 + dx, where a ∈ F * q and b, c, d ∈ F q . We consider when N r, =…”

“…The following theorem is a consequence of [17,14,6] and gives the conditions for a curve defined over a field of characteristic 2 associated with a Jacobian that has 2-rank 1 to exist. Cardona-Pujolas-Nart in [6] showed that such a 2-rank 1 curve will be of the form given in Theorem 1. (Our statement combines Lemma 2.1, Theorem 2.9 part (M) and Corollary 2.17 of [17], as it appears in [18].)…”