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.