Modular polynomials for genus 2

TL;DR

This paper explores the generalization of modular polynomials from elliptic curves to genus 2 curves, establishing their properties and providing improved computational methods.

Contribution

It introduces new properties of genus 2 modular polynomials and presents an enhanced explicit computation technique.

Findings

Proved various properties of genus 2 modular polynomials

Developed an improved explicit computation method

Extended the application scope of modular polynomials to genus 2

Abstract

Modular polynomials are an important tool in many algorithms involving elliptic curves. In this article we investigate their generalization to the genus 2 case following pioneering work by Gaudry and Dupont. We prove various properties of these genus 2 modular polynomials and give an improved way to explicitly compute them.