We construct Weil numbers corresponding to genus-2 curves with p-rank 1 over the finite field Fp2 of p2 elements. The corresponding curves can be constructed using explicit CM constructions. In one of our algorithms, the group of Fp2-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 Fp2 out of necessity: we show that curves of p-rank 1 over Fp for large p cannot be efficiently constructed using explicit CM constructions. © 2010 Elsevier Inc.
Hitt O’Connor, L., McGuire, G., Naehrig, M., & Streng, M. (2011). A CM construction for curves of genus 2 with p-rank 1. Journal of Number Theory, 131(5), 920–935. https://doi.org/10.1016/j.jnt.2010.05.002