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

We construct Weil numbers corresponding to genus-2 curves with p-rank 1 over the finite field F_p² of p² elements. The corresponding curves can be constructed using explicit CM constructions. In one of our algorithms, the group of F_p²-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 F_p² out of necessity: we show that curves of p-rank 1 over F_p for large p cannot be efficiently constructed using explicit CM constructions.