Determine the unresolved 3-primary component of cyclic supersingular pairings

Determine whether the 3-primary component of the pairings e_m(P,R) and e_m(P,Q) for supersingular abelian surfaces of type b=1 is trivial, thereby deciding whether e_m(P,R) has exact order m/3 when 3 divides m.

Background

For a supersingular abelian surface of type b=1, the exponent is m=p2-p+1. When p is congruent to 2 modulo 3, the integer m is divisible by 3, and the eigenspaces used to construct the relevant Weil pairings are not distinct modulo 3. The paper proves that the principal pairing terms have order m/3 or m, but does not determine their 3-primary component.

The authors report experimental evidence that the 3-primary component is trivial, which would imply that the relevant pairing has exact order m/3. Establishing this rigorously would sharpen the pairing structure used in conclusive supersingularity verification, although the stated verification corollary does not depend on resolving this ambiguity.

References

The 3-part is not determined: modulo 3 the eigenvalues ±1, ±p collapse in pairs, and e_3 need not vanish between the two resulting order 3 eigengroups. Experiments suggest the 3-part of e_m(P,R) and e_m(P,Q) is trivial, i.e., that e_m(P,R) has order exactly m/3; Cref{cor:conclusive-testing-cyclic} does not depend on this.

— Supersingularity and Superspeciality Verification of Abelian Surfaces  (2610.01924 - Santos et al., 1 Oct 2026) in Section 6, Lemma “The case b = 1,” proof and footnote