Units of hyperelliptic curves over $\mathbb{F}_2$
Justin Chen, Vishal Muthuvel
TL;DR
This work investigates unit groups of coordinate rings $R=\mathbb{F}_2[x,y]/(y^2+gy+h)$ arising from affine plane curves over $\mathbb{F}_2$ with a 2-to-1 map to the line. The authors classify such rings up to ambient automorphisms into three minimal types, relate units to representations of $1$ by the binary quadratic form $Q(a,b)=a^2+abg+b^2h$, and establish degree-based constraints that render Types 1 and 2 always trivial while Type 3 remains conjecturally nontrivial. A fundamental unit concept is developed, showing $R^\times$ is either trivial or infinite cyclic in Type 3, with symmetry operations guaranteeing that all units are powers of a fundamental unit; an algorithm based on Gröbner bases computes these units in high degrees and yields extensive computational evidence, including accumulative degeneracy patterns and growth of fundamental unit degrees. Overall, the paper provides a concrete framework for understanding unit groups in this class of rings, proposes a precise Type 3 conjecture, and delivers practical methods for computing units in large-degree cases, with potential implications for the arithmetic of hyperelliptic curves over $\mathbb{F}_2$.
Abstract
We study unit groups of rings of the form $\mathbb{F}_2[x,y]/(y^2 + gy + h)$, for $g, h \in \mathbb{F}_2[x]$ -- in particular, the question of (non)triviality of such unit groups. Up to automorphisms of $\mathbb{F}_2[x,y]$ we classify such rings into 3 distinct types. For 2 of the types we show that the unit group is always trivial, and conjecture that the unit group is always nontrivial for the 3rd type. We provide support for this conjecture both theoretically and computationally, via an algorithm that has been used to compute units in large degrees.