  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.

Algebraic Geometry