Algebraic solutions of linear differential equations: an arithmetic approach

TL;DR

This paper explores an arithmetic approach to determine whether solutions of linear differential equations with rational function coefficients are algebraic, highlighting connections to deep conjectures like Grothendieck-Katz p-curvature.

Contribution

It introduces an elementary local-global arithmetic method for analyzing algebraic solutions of linear differential equations, rooted in Grothendieck's framework.

Findings

Illustrates the approach with motivating examples from mathematics

Links the problem to the Grothendieck-Katz p-curvature conjecture

Provides insights into the algebraic nature of solutions

Abstract

Given a linear differential equation with coefficients in $\mathbb{Q}(x)$, an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic. After presenting motivating examples coming from various branches of mathematics, we advertise in an elementary way a beautiful local-global arithmetic approach to these questions, initiated by Grothendieck in the late sixties. This approach has deep ramifications and leads to the still unsolved Grothendieck-Katz $p$-curvature conjecture.