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

Laura Hitt O'Connor, Gary McGuire, Michael Naehrig, and Marco Streng

Abstract

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.

Details

Publication typeArticle
Published inJournal of Number Theory
PublisherElsevier
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