Hyperelliptic curves with zero Hasse-Witt matrix

Abstract

We consider hyperelliptic curves in characteristic p>2 for which the Hasse-Witt matrix is the zero matrix. For such curves, we establish an upper bound on the genus g, namely g≤(p-1)/2. For g=(p-1)/2, we establish the fact that up to isomorphism, there is precisely one such curve, namely the one given by the equation y2=xp-x. We determine all hyeprelliptic curves of the form y2=xn-x or y2=xn-1 for which the Hasse-Witt matrix is the zero matrix.