UNITS OF HYPERELLIPTIC CURVES OVER F2

We study unit groups of rings of the form 𝔽2[x,y]∕(y2+gy+h), for g,h∈𝔽2[x] — in particular, the question of when such unit groups are trivial. Up to automorphisms of 𝔽2[x,y] we classify such rings into three types. For two of the types we show that the unit group is always trivial, and conjecture that the unit group is always nontrivial for the third type. We provide support for this conjecture both theoretically and computationally, via an algorithm that has been used to compute units in large degrees.