A CM construction of 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 F_p^2 of p^2 elements. The corresponding curves can be constructed using explicit CM constructions. In one of our algorithms, the group of F_p^2-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_p^2 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