Characterize the α_i for fourth-order operators: uniqueness and formula for α_3

Prove that, after an appropriate algebraic change of the parameter φ fixing the MUM point, a fourth-order Calabi–Yau type differential operator has exactly one independent non-zero constant α_3 in U_p(0), and that α_3 equals K ζ_p(3) with K ∈ Q independent of p and determined by the operator’s rational structure.

Background

In the Frobenius matrix U_p(0) for order-4 operators, the constants α_i are constrained by the polarization but not fully determined. The authors record a conjecture specifying the structure: only α_3 remains as an independent non-zero parameter, and it has the form α_3 = K ζ_p(3).

This links the arithmetic of U_p(0) to a rational number K arising from the rational structure associated with L, and thus ties the constants in U_p(0) to geometric data.

References

For example, for operators with b=4, it is conjectured that (subject to a judicious choice of the coordinate φ, see ref.), there is only one independent non-zero coefficient α_3. This takes the form α_3 = K ζ_p(3), where K ∈ Q is a constant independent of the prime p and determined by the rational structure associated to L.

— Solutions of Calabi-Yau Differential Operators as Truncated p-adic Series and Efficient Computation of Zeta Functions  (2604.01191 - Kuusela et al., 1 Apr 2026) in Section 2.2 (following equation (2.17))