A CM construction for curves of genus 2 with p-rank 1

Abstract: We construct Weil numbers corresponding to genus-2 curves with p-rank 1 over the finite field of elements. The corresponding curves can be constructed using explicit CM constructions. In one of our algorithms, the group of -valued points of the Jacobian has prime order, while another allows for a prescribed embedding degree with respect to a subgroup of prescribed order. The curves are defined over out of necessity: we show that curves of p-rank 1 over for large p cannot be efficiently constructed using explicit CM constructions. [Copyright &y& Elsevier]