An explicit Construction of the Jacobian of a nonsingular curve and its group law

Abstract

This paper is devoted to constructing an explicit efficient representation for the Jacobian variety of a nonsingular curve of genus greater than 1, and its group law. We describe an algorithm for executing the group law on the Jacobian elements, consisting of several steps. Finally we exhibit two examples. The first example exhibits an explicit formulation for the Jacobian of a well known non singular elliptic curve of genus 1, that is, the Weierstrass curve. The second example introduces an explicit computation of the group law for a non singular curve of genus 3, $C : \omega_2^4 = \omega_1 (\omega_1-\omega_0)(\omega_1-2\omega_0)(\omega_1-3\omega_0)$, where $\omega_0, \omega_1, \omega_2$ are homogeneous coordinates of the projective plane $\mathbb{P}^2$. These examples are highly detailed for the convenience of the reader.