Arithmetic nature of the Eichler-type integral

Characterize the arithmetic nature of the Eichler-type integral arising from the weight-four determinant-character form $\Delta W^2/v$, including its Eisenstein and cuspidal components on the genus-one twist cover.

Background

The paper proves that the derivative of the generating function is expressible through a depth-one quasimodular form, while the generating function itself requires an additional integral involving the rational obstruction Δ(t)=(13t−1)/(30t(t−1)2)\Delta(t)=(13t-1)/(30t(t-1)^2).

After rewriting this remaining term as an Eichler-type integral of the weight-four form ΔW2/v\Delta W^2/v, the authors do not determine its finer arithmetic decomposition. In particular, the Eisenstein and cuspidal contributions on the genus-one modular twist cover remain unresolved.

References

The remaining transcendental in eq:phiid is the Eichler-type integral of the weight-four, determinant-character form $\Delta\,W2/v$ (via $d t=W\,d q/q$); its arithmetic nature---its Eisenstein and cuspidal components on the twist cover of eq:twist---we leave open.

— Lattice Green's function of the hyperkagome lattice: modular uniformization at level 30 from an orthogonal differential Galois group  (2608.28141 - Nasr et al., 28 Aug 2026) in Section 5, Section~\ref{sec:phiobs}, immediately following Theorem~\ref{thm:phi}