Special-function interpretation of quadratic-twist cross-relations

Investigate whether the cross-relations between the critical values of the weight-4 modular form \(\phi(\tau)=\eta(\tau)^2\eta(2\tau)^2\eta(3\tau)^2\eta(6\tau)^2\) and its quadratic twist \(\psi=\phi\otimes\chi_{-3}\) can be understood directly through special-function representations, specifically by determining whether twisting by \(\chi_{-3}\) corresponds to a transformation of the associated elliptic-integral parametrizations or by constructing independent Bessel-moment representations for the critical values of \(\psi\).

Background

The paper establishes cross-relations between critical values of the level-6 modular form ϕ\phi and those of its quadratic twist ψ=ϕ⊗χ−3\psi=\phi\otimes\chi_{-3} using period polynomials, modular symbols, and a conductor-3 cusp geometry. Earlier work provides Bessel-moment evaluations for critical values of ϕ\phi, but the paper does not supply corresponding elliptic-integral or Bessel-moment representations for the critical values of ψ\psi.

The unresolved problem is to identify a direct special-function counterpart of the modular quadratic twist. Such a result could convert the established cross-relations into explicit identities between distinct elliptic-integral or Bessel moments, thereby extending the known connections among modular LL-values, elliptic integrals, Bessel moments, and Feynman-integral structures.

References

This leads naturally to the question of whether the cross-relations in eq:quadratic-twist-cross-relations can also be understood directly on the special-function side. Since the twist is governed by the character \chi_{-3}, one may ask whether the passage from \phi to \psi is reflected in a corresponding transformation of the associated elliptic-integral parametrizations. Alternatively, independent Bessel-moment representations for the critical values of \psi would turn the cross-relations above into explicit identities between different Bessel moments.

We do not pursue these questions further here. The purpose of the present discussion is to point out that the period-polynomial and modular-symbol relations obtained above fit naturally into the existing theory connecting modular L-values with elliptic and Bessel moments. A systematic investigation of the special-function representations associated with the quadratic twist \psi is left for future research.

eq:quadratic-twist-cross-relations:

L(ψ,3)=4π33L(ϕ,2),L(ϕ,3)=π310L(ψ,2).L(\psi,3)=\frac{4\pi}{3\sqrt{3}}L(\phi,2), \qquad L(\phi,3)=\frac{\pi\sqrt{3}}{10}L(\psi,2).

— Period Relations for Theta Products and Elliptic Integral Moments  (2608.26514 - Bao, 27 Aug 2026) in Section 5, subsection “Connection with elliptic and Bessel moments” (Section 5.1, immediately following Theorem 5.1)