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

TL;DR

This paper develops explicit CM constructions for genus-2 curves with p-rank 1 over finite fields, enabling the creation of curves with specific properties such as prime order Jacobians and prescribed embedding degrees.

Contribution

It introduces new algorithms for constructing genus-2 curves with p-rank 1 over _{p^2} using explicit CM methods, including cases with prime Jacobian order and prescribed embedding degrees.

Findings

Curves of p-rank 1 over _p are hard to construct via explicit CM for large p.

Constructed curves over _{p^2} can have prime Jacobian order.

Algorithms allow for prescribed embedding degrees in the constructed curves.

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.