Computing unit groups of curves

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 necessary in order 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.

Introduction

Among the invariants of a commutative ring, the group of units is one of the most fundamental. However, explicit computation of this group is difficult, and even its structure remains mysterious in general (Fuchs, 1960). To date, most progress has centered on rings of integers of algebraic number fields, or localizations thereof, driven by a need for practical algorithms in computational number theory (Cohen, 1993). These results rely fundamentally on Dirichlet's unit theorem, which describes the group of units, modulo torsion, of a number field as a free abelian group of finite rank specified by simple invariants of the number field.

An analogous theorem, proved independently by Rosenlicht and Samuel (Rosenlicht, 1957; Samuel, 1966), states that for a finitely generated domain over an algebraically closed field, the group of units, modulo scalars, is free abelian of finite rank. In contrast to the number field case, no formula for the rank is known. However, there is still interest in understanding the unit group: for a very affine variety, generators for the unit group of the coordinate ring yield an embedding of the variety into its so-called intrinsic torus (Maclagan and Sturmfels, 2015). In tropical geometry, this embedding of a very affine variety into its intrinsic torus realizes its intrinsic tropicalization, from which all other tropicalizations can be recovered. Computing the intrinsic tropicalization is difficult, though, because one must first compute the unit group.

In this work we describe effective methods for computing unit groups of smooth very affine curves of low genus. Our methods rely on divisor theory for projective varieties: we embed the unit group of a very affine variety into the Weil divisor group of the projective closure, and study the cokernel of this embedding as a subgroup of the divisor class group. This allows us to give algorithms for computing unit groups of rational normal curves and elliptic curves:

Theorem 1.1

Let $\bar{C}\subseteq {\mathbb{P}}_k^n$ be a rational normal curve over an algebraically closed field k, given parametrically as the image of a map ${\mathbb{P}}_k^1\hookrightarrow {\mathbb{P}}_k^n$. Let $C:=\bar{C}\cap {\mathbb{T}}^n$ be the corresponding very affine curve, with coordinate ring R. Then Algorithm 5.4 correctly computes a $\mathbb{Z}$-basis of $R^{⁎}/k^{⁎}$.

Theorem 1.2

Let $k=\bar{\mathbb{Q}}$, let $\bar{E}\subseteq {\mathbb{P}}_k^2$ be an elliptic curve, and let $E:=\bar{E}\cap {\mathbb{T}}^2$ be the corresponding very affine elliptic curve with coordinate ring R. Then Algorithm 6.11 correctly computes a $\mathbb{Z}$-basis of $R^{⁎}/k^{⁎}$.

We briefly describe the structure of the paper. The basics of unit groups and intrinsic tropicalizations are discussed in Section 2. In Section 3 we interpret the problem of computing units via the geometry of boundary divisors, and give an algorithm for interpolating divisors of rational functions to Laurent polynomials. We consider two families of plane curves in Section 4, namely Fermat curves and conics. Section 5 deals with rational normal curves in parametric form. Finally, we discuss elliptic curves in Section 6.

Many of our algorithms have been implemented in Macaulay2 (Grayson and Stillman, 2018), Singular (Decker et al., 2018), or Sage (The Sage Developers, 2018). Our code for the examples in this paper can be found at: https://github.com/leonyz/unitgroups/.

Leon Zhang and Sameera Vemulapalli would like to thank the Max Planck Institute for Mathematics in the Sciences for its hospitality while working on this project. Leon Zhang was supported by a National Science Foundation Graduate Research Fellowship under Grant No. 1752814.

The authors thank Bernd Sturmfels for suggesting and advising this project. We would also like to thank Chris Eur and Martin Helmer for helpful discussions, Yue Ren for generous and thoughtful help in computing tropicalizations in Singular, and Bjorn Poonen and Ronald van Luijk for their expertise and guidance.