Counting isomorphism classes of pointed hyperelliptic curves of genus 4 over finite fields with even characteristic. (English) Zbl 1206.14044
Any hyperelliptic curve \(C\) of genus \(g\), having a rational Weierstrass point, admits a non-singular affine model of the type: \(v^2+h(u)v=f(u)\), where \(h(u),f(u)\) are polynomials of respective degree \(\deg h\leq g\), \(\deg f=2g+1\). For \(C\) given by such a model, P. Lockhart characterized the isomorphism classes of pointed curves \((C,P)\), where \(P\) is the Weierstrass point of \(C\) at infinity, in terms of explicit plane coordinate transformations preserving the shape of the model [P. Lockhart, Trans. Am. Math. Soc. 342, No. 2, 729–752 (1994; Zbl 0815.11031)].
The paper under review is devoted to counting the number of isomorphism classes of pointed hyperelliptic curves of genus \(4\) over finite fields of characteristic \(2\), by counting the number of Lockhart’s transformations. For finite fields of odd characteristic the number of (pointed and non-pointed) hyperelliptic curves of arbitrary genus \(g\) is counted in [the reviewer, Adv. Math. 221, No. 3, 774–787 (2009; Zbl 1214.11078)].
The paper under review is devoted to counting the number of isomorphism classes of pointed hyperelliptic curves of genus \(4\) over finite fields of characteristic \(2\), by counting the number of Lockhart’s transformations. For finite fields of odd characteristic the number of (pointed and non-pointed) hyperelliptic curves of arbitrary genus \(g\) is counted in [the reviewer, Adv. Math. 221, No. 3, 774–787 (2009; Zbl 1214.11078)].
Reviewer: Enric Nart Viñals (Barcelona)
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