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Cais, Ellenberg and Zureick-Brown recently observed that over finite fieldsof characteristic two, all sufficiently general smooth plane projective curvesof a given odd degree admit a non-trivial rational 2-torsion point on theirJacobian. We extend their observation to curves given by Laurent polynomialswith a fixed Newton polygon, provided that the polygon satisfies a certaincombinatorial property. We also show that in each of these cases, thesufficiently general condition is implied by being ordinary. Our treatmentincludes many classical families, such as hyperelliptic curves of odd genus and$C_{a,b}$ curves. In the hyperelliptic case, we provide alternative proofsusing an explicit description of the 2-torsion subgroup.

Curves in characteristic 2 with non-trivial 2-torsion