Dice Question Streamline Icon: https://streamlinehq.com

Local algebraicity vs. monodromy at singular points for logarithmic flat bundles

Determine whether, for a logarithmic flat bundle (E,∇:E→E⊗Ω^1_X(log D)) on a smooth projective complex curve X with reduced divisor D and an integral formal flat section s at a point x∈D, either s is algebraic or the bundle (E,∇) has infinite local monodromy at x.

Information Square Streamline Icon: https://streamlinehq.com

Background

This local prediction refines the main arithmetic/algebraicity conjecture to behavior at singular points, proposing a dichotomy between algebraicity of an integral formal flat section and infinite local monodromy.

The authors verify this conjecture for hypergeometric functions by combining results of Christol and Beukers–Heckman, but the general case remains open.

References

Conjecture Let X/\mathbb{C} be a smooth projective curve, D\subset X a reduced effective divisor, and (\mathscr{E}, \nabla: \mathscr{E}\to \mathscr{E}\otimes \Omega1_X(\log D)) a logarithmic flat bundle on (X, D). Let x\in D be a point and let s be an integral formal flat section to (\mathscr{E}, \nabla) at x. Then either s is algebraic, or (\mathscr{E}, \nabla) has infinite monodromy at x.

— Algebraicity and integrality of solutions to differential equations (2501.13175 - Lam et al., 22 Jan 2025) in Conjecture, Section “Singular points”