Higher-degree elliptic subcovers of real band-three harmonics

Characterize the real degree-three spherical harmonics whose associated sextic curve has an elliptic subcover of degree higher than two, and determine the corresponding constraints on their Maxwell-axis configurations.

Background

For a real harmonic of degree three, the associated binary sextic defines a genus-two curve. The paper characterizes the degree-two elliptic-subcover locus as the achiral locus, equivalently as the vanishing locus of the skew invariant, and relates this to coplanar or bisector-type configurations of the three Maxwell axes.

The geometry of real band-three harmonics associated with elliptic subcovers of higher degree remains unresolved. In particular, the paper does not determine which real harmonics lie on such loci or what geometric restrictions those loci impose on their Maxwell axes.

References

Which real harmonics of degree three have a sextic whose curve has an elliptic subcover of higher degree, and what configuration of their Maxwell axes this imposes, is open; unlike the degree-two case, no reflection is involved, and no anatomical meaning should be assigned to such loci before empirical validation.

— Invariant Shape Analysis of Surfaces with Spherical Topology  (2609.39567 - Shaska et al., 30 Sep 2026) in Section 10, Discussion (final section; paragraph beginning “Three problems remain open”)