Resolve the descent obstruction over the base field

Determine whether every geometric level admitting a separable model is realized by a cyclic cover and a separable model defined over the original field k_0, or characterize and resolve the remaining obstruction to obtaining a Kummer model over k_0.

Background

The geometric algorithm works after extending the base field by adjoining relevant Weierstrass points and roots of unity. The arithmetic discussion analyzes when the computation can instead be carried out over k_0, but the paper does not establish that every geometrically realizable separable level descends to that field. In particular, after accounting for the possible conic obstruction and subgroup descent issues, a remaining obstruction to constructing a Kummer equation over k_0 is left untreated.

References

Two questions are left open. Levels realised only by cyclic covers that admit no separable model are outside \cref{thm:complete}, and \cref{rem:incomplete} gives branch data, $n=4$ with $(l_j)=(1,1,3,3)$ in genus three, for which the reduction to Weierstrass points fails; deciding those levels needs a different search. Over $k_0$, the run returns the geometric answer of \cref{thm:complete} exactly when every level with a separable model is realised by a cover and a model defined over $k_0$, and \cref{rem:descent} leaves the remaining obstruction to a Kummer model over the field untreated.

— Deciding superellipticity and computing the Weierstrass normal form  (2609.00672 - Shaska, 1 Sep 2026) in Section Implementation and timings, final paragraph