Extend affine quotient-data methods to wild and non-exceptional genus-one covers

Determine whether an appropriate affine quotient datum continues to separate the geometric form of a genus-one cover from the arithmetic carried by Frobenius relative to the marked affine symmetry in (i) the genuinely wild genus-one theory in characteristics 2 and 3 and (ii) genus-one monodromy configurations beyond the exceptional affine setting.

Background

The paper completely classifies tame exceptional covers of the projective line whose geometric Galois closure has genus one by means of Frobenius-stable affine elliptic quotient data. These data encode the elliptic quotient, the cyclic affine symmetry and its shift, and the Frobenius action on the translation kernel.

The authors identify two directions not covered by the classification: genuinely wild genus-one covers in characteristics 2 and 3, and genus-one monodromy configurations that lie outside the exceptional affine setting. The unresolved issue is whether the same conceptual separation between geometric form and Frobenius-based arithmetic survives in either broader context.

References

Two natural extensions remain. The first is the genuinely wild genus-one theory in characteristics $2$ and $3$. The second is the fixed-field classification of genus-one monodromy configurations beyond the exceptional affine setting. In both directions, the central question is whether an appropriate affine quotient datum continues to separate the geometric form of the cover from the arithmetic carried by Frobenius relative to the marked affine symmetry.

— Exceptional covers of the projective line with genus-one Galois closure: arithmetic forms over finite fields  (2608.27255 - Fan, 27 Aug 2026) in Section 6, Outlook (Section 6 is labeled “Outlook” in the source)