Paper 2022/1107

Projective Geometry of Hessian Elliptic Curves and Genus 2 Triple Covers of Cubics

Abstract

The existence of finite maps from hyperelliptic curves to elliptic curves has been studied for more than a century and their existence has been related to isogenies between a product of elliptic curves and their Jacobian surface. Such finite covers, sometimes named gluing maps have recently appeared in cryptography in the context of genus 2 isogenies and more spectacularly, in the work of Castryck and Decru about the cryptanalysis of SIKE. Computation methods include the use of algebraic theta functions or correspondences such as Richelot isogenies or degree 3 analogues. This article aims at giving geometric meaning to the gluing morphism from a product of elliptic curves $E_1 \times E_2$ to a genus 2 Jacobian when it is a degree (3, 3) isogeny. An explicit (uni)versal family and an algorithm were previously provided in the literature (Bröker-Howe-Lauter-Stevenhagen) and a similar special case was studied by Kuwata. We provide an alternative construction of the universal family using concepts from classical algebraic and projective geometry. The family of genus 2 curves which are triple covers of 2 elliptic curves with a level 3 structure arises as a correspondence given by a polarity relation. The construction does not provide closed formulas for the final curves equations and morphisms. However, an alternative algorithm based on the geometric construction is proposed for computation on finite fields. It relies only on elementary operations without requiring polynomial roots and computes the equation of the genus 2 curves and morphisms in all cases.

Note: Revision 13 Oct 2022 Revamped algorithm without any square root (better complexity) Additional formulas (2nd coordinate of ramification points, bilinear correspondence in Weierstrass form) Style and typo corrections, expanded introduction