Fast Enumeration of Superspecial Hyperelliptic Curves of Genus 4 with Automorphism Group $$V_4$$

Abstract

In this paper, we examine superspecial genus-2 curves C :As a main result, we show that the difference between any two elements in {0, 1, λ, µ, ν} is a square in F p 2 . Moreover, we show that C is maximal or minimal over F p 2 without taking its F p 2 -form (we also give a criterion in terms of p that tells whether C is maximal or minimal). As these applications, we study the maximality of superspecial hyperelliptic curves of genus 3 and 4 whose automorphism groups contain Z/2Z × Z/2Z. Rosenhain forms of genus-2 curveAny genus-2 curve has just 6 Weierstrass points, and we can consider an isomorphism C ∼ = C λ,µ,ν which maps three of them to 0, 1 and ∞.Definition 2.1. Given a genus-2 curve C, we say thatis a Rosenhain form of C when there exists an isomorphism C ∼ = C λ,µ,ν over the algebraic closure of K. Then the values λ, µ and ν are called Rosenhain invariants of C λ,µ,ν .