The Four Faces of Hyperelliptic curves

Abstract

In this thesis we will look at elliptic and hyperelliptic curves. There are three abelian groups that are isomorphic to hyperelliptic curves. The Jacobian of hyperelliptic curves, the ideal class group and the form class group, will all be defined and given abelian group structure. We will give an algorithm for point addition and point doubling done exclusively in the jacobian of the curve. We will end the thesis with proving that there exists an isomorphism between the form class group and the ideal class group.