Constructing all Genus 2 Curves with Supersingular Jacobian

Abstract

L. Moret-Bailly constructed families $\mathfrak{C}\to {\mathbb{P}}^1$ of genus 2 curves with supersingular jacobian. In this paper we first classify the reducible fibers of a Moret-Bailly family using linear algebra over a quaternion algebra. The main result is an algorithm that exploits properties of two reducible fibers to compute a hyperelliptic model for any irreducible fiber of a Moret-Bailly family.

28 pages. MAGMA Code is available on GitHub