Constructing all Genus 2 Curves with Supersingular Jacobian

TL;DR

This paper provides a classification of reducible fibers in Moret-Bailly families of genus 2 curves with supersingular Jacobians and introduces an algorithm to explicitly construct hyperelliptic models for these fibers.

Contribution

It offers a classification of reducible fibers using quaternion algebra and presents an algorithm to construct hyperelliptic models for irreducible fibers in these families.

Findings

Classification of reducible fibers via quaternion algebra

Algorithm for hyperelliptic model construction

Explicit models for all genus 2 curves with supersingular Jacobians

Abstract

L. Moret-Bailly constructed families $\mathfrak{C}\rightarrow \mathbb{P}^1$ 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.