Hypergeometric configuration spaces and higher-order modular differential equations

Investigate whether the general theory of hypergeometric functions and configuration spaces can provide a framework for studying higher-order or multivariable modular differential equations.

Background

The paper derives connection matrices for the Kaneko–Zagier differential equation by transforming it into Gauss’s hypergeometric equation. In a supplementary remark, the authors interpret trigonometric identities arising from Ptolemy’s theorem as Plücker relations on the Grassmannian configuration space of four points, linking the algebraic calculations to the geometric structure of hypergeometric differential equations.

The authors identify a speculative direction extending this relationship beyond the second-order, single-variable setting considered in the paper. The unresolved issue is whether the broader theory of hypergeometric functions and configuration spaces can yield a systematic framework for higher-order or multivariable modular differential equations. No resolution of this question is provided.

References

Although the KZ equation is a canonical example of modular differential equations, the appearance of such an algebraic structure raises an interesting, albeit speculative, question: whether the general theory of hypergeometric functions and configuration spaces---developed by Gelfand, Aomoto, Yoshida, and others---might one day provide a framework for exploring higher-order or multivariable modular differential equations.

— An Algebraic Proof of the Non-Modularity of Solutions to the Canonical Modular Differential Equation  (2610.05668 - Sakai et al., 5 Oct 2026) in Remark following Theorem 3.4, Section 3 (Algebraic Criterion for Simultaneous Triangularization)