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Arithmetic/algebraicity criterion for formal flat sections of linear differential equations

Establish, for a flat vector bundle (E,∇) on a smooth R-scheme X over a finitely generated Z-algebra R⊂C, a point x∈X(C) and an initial condition v∈E_x, the equivalence between (i) algebraicity of the formal flat section s through v; (ii) integrality of s; and (iii) ω(p)-integrality of s.

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Background

This conjecture specializes the general foliation conjecture to flat bundles (linear ODE), proposing that sporadic algebraic flat sections can be characterized arithmetically through integrality of Taylor expansions.

In the linear setting it is stronger than Grothendieck–Katz p-curvature and relates to variants attributed to André and Christol, but the full equivalence remains open.

References

Conjecture Let R\subset\mathbb{C} be a finitely-generated \mathbb{Z}-algebra, X a smooth R-scheme, and (\mathscr{E}, \nabla) a flat vector bundle on X/R. Let x\in X(\mathbb{C}) be a point, and fix v\in \mathscr{E}_x. Then the formal flat section to \mathscr{E} through v is algebraic if and only if it is integral, if and only if it is \omega(p)-integral.

— Algebraicity and integrality of solutions to differential equations (2501.13175 - Lam et al., 22 Jan 2025) in Conjecture, Part I (Linear differential equations), Section “Main results for linear ODE”