A ug 2 00 4 Zeta functions of supersingular curves of genus 2

This paper was motivated by the problem of determining what isogeny classes of abelian surfaces over a finite field k contain jacobians. In [MN] we performed a numerical exploration of this problem, that led to several conjectures. We present in this paper a complete answer for supersingular surfaces in characteristic 2 (section 5). We deal with this problem in a direct way by computing explicitly the zeta function of all supersingular curves of genus two (section 4). 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. We base our work on the ideas of van der Geer and van der Vlugt [vdGvdV1], [vdGvdV2], who expressed the number of points of a supersingular curve of genus two in terms of certain invariants. In section 2 we compute explicitly these invariants in terms of the coefficients of a defining equation and in section 3 we compute the number of points of the curve over the quadratic extension in terms of objects defined over k.