Curves of genus two over fields of even characteristic

Abstract

In this paper we classify curves of genus two over a perfect field k of characteristic two. We find rational models of curves with a given arithmetic structure for the ramification divisor and we give necessary and sufficient conditions for two models of the same type to be k-isomorphic. As a consequence, we obtain an explicit formula for the number of k-isomorphism classes of curves of genus two over a finite field. Moreover, we prove that the field of moduli of any curve coincides with its field of definition, by exhibiting rational models of curves with any prescribed value of their Igusa invariants. Finally, we use cohomological methods to find, for each rational model, an explicit description of its twists. In this way, we obtain a parameterization of all k-isomorphism classes of curves of genus two in terms of geometric and arithmetic invariants.