  We determine what isogeny classes of supersingular abelian surfaces over a
finite field k of characteristic 2 contain jacobians. We deal with this problem
in a direct way by computing explicitly the zeta function of all supersingular
curves of genus 2. Our procedure is constructive, so that we are able to
exhibit curves with prescribed zeta function and to count the number of curves,
up to k-isomorphism, leading to the same zeta function.
