Computing Unit Groups of Curves

TL;DR

This paper introduces practical algorithms for computing the unit groups of smooth low-genus curves, which are crucial for applications in tropical geometry, using divisor theory and algebraic number theory methods.

Contribution

It provides new algorithms for computing unit groups of curves, combining divisor interpolation and algebraic number theory techniques.

Findings

Algorithms successfully compute unit groups for low-genus curves.

Approach is effective for rational and elliptic curves.

Facilitates intrinsic tropicalizations in tropical geometry.

Abstract

The group of units modulo constants of an affine variety over an algebraically closed field is free abelian of finite rank. Computing this group is difficult but of fundamental importance in tropical geometry, where it is desirable to realize intrinsic tropicalizations. We present practical algorithms for computing unit groups of smooth curves of low genus. Our approach is rooted in divisor theory, based on interpolation in the case of rational curves and on methods from algebraic number theory in the case of elliptic curves.