The arithmetic of genus two curves

TL;DR

This paper explores the fundamental properties, moduli spaces, and arithmetic of genus 2 curves, emphasizing their significance in cryptography and providing a comprehensive overview suitable for graduate students.

Contribution

It offers a detailed survey of genus 2 curves, including moduli spaces, Humbert surfaces, modular polynomials, and their cryptographic applications, filling a gap in educational resources.

Findings

Overview of genus 2 curve properties

Description of moduli spaces and Humbert surfaces

Applications to cryptography and algorithm design

Abstract

Genus 2 curves have been an object of much mathematical interest since eighteenth century and continued interest to date. They have become an important tool in many algorithms in cryptographic applications, such as factoring large numbers, hyperelliptic curve cryptography, etc. Choosing genus 2 curves suitable for such applications is an important step of such algorithms. In existing algorithms often such curves are chosen using equations of moduli spaces of curves with decomposable Jacobians or Humbert surfaces. In these lectures we will cover basic properties of genus 2 curves, moduli spaces of (n,n)-decomposable Jacobians and Humbert surfaces, modular polynomials of genus 2, Kummer surfaces, theta-functions and the arithmetic on the Jacobians of genus 2, and their applications to cryptography. The lectures are intended for graduate students in algebra, cryptography, and related areas.