Construction of CM Picard curves

Abstract

Abstract. In this article we generalize the CM method for elliptic and hyperelliptic curves to Picard curves. We describe the algorithm in detail and discuss the results of our implementation.

A Generalized Brezing-Weng Algorithm for Constructing Pairing-Friendly Ordinary Abelian Varieties

“…In dimension g = 2 we compute the Igusa class polynomials, which are three polynomials in Q[x] whose roots are the Igusa invariants of genus 2 curves over Q whose Jacobians have CM by O K . Methods for g = 3 are analogous but have only been developed for fields K containing i or ζ 3 [23,15]. Methods for g ≥ 4 are completely undeveloped.…”

“…Abelian varieties with CM by Q(ζ 9 ) are Jacobians of Picard curves of the form y 3 = x 4 + ax [15]. We note that since these curves are not hyperelliptic, for any q-Weil number π ∈ Z[ζ 9 ] there is a curve C/F q whose Jacobian has Frobenius element either π or −π.…”

Women in Numbers Europe

“…We end this section by considering Example 3 from Section 5 of Koike and Weng (2005), wherein Koike and Weng study Picard curves with CM. We show that the curve in the aforementioned example has bad reduction to characteristic p D 5, and that the stable reduction consists of an elliptic curve and a curve of genus 2.…”

Index Calculus in Class Groups of Non-hyperelliptic Curves of Genus Three

“…Particular families of such curves proposed for cryptography include Picard curves [10,17,4,26] and more generally C 3,4 -curves [2,3]. For these families of curves, effort has been put into providing efficient construction means and explicit formulae for computations in the degree 0 class group.…”