On the Rosenhain forms of superspecial curves of genus two

In this paper, we examine superspecial genus-2 curves C:y2=x(x−1)(x−λ)(x−μ)(x−ν) in odd characteristic p. As a main result, we show that the difference between any two elements in {0,1,λ,μ,ν} is a square in Fp2. Moreover, we show that C is maximal or minimal over Fp2 without taking its Fp2-form (we give an explicit criterion in terms of p that tells whether C is maximal or minimal). As an application, we also study the maximality of superspecial hyperelliptic curves of genera 3 and 4 whose automorphism groups contain Z/2Z×Z/2Z.