A characterization of the Artin–Mumford curve

Let MM be the Artin–Mumford curve over the finite prime field FpFp with p>2p>2. By a result of Valentini and Madan, AutFp(M)≅HAutFp(M)≅H with H=(Cp×Cp)⋊Dp−1H=(Cp×Cp)⋊Dp−1. We prove that if XX is an algebraic curve of genus g=(p−1)2g=(p−1)2 defined over FpFp such that AutFp(X)AutFp(X) contains a subgroup isomorphic to H   then XX is birationally equivalent over FpFp to the Artin–Mumford curve MM.