Differential equations with 2-term recursion (Q1384678)

A well known Grothendieck conjecture says that a linear differential equation \(L\) over \({\mathbb Q}(x)\) has a fundamental system of algebraic solutions (hence a finite Galois group) if and only if the reduced equation over \({\mathbb F}_{p}(x)\) has a fundamental system of rational solutions for almost all \(p\). The aim of this paper is to study this conjecture when there exists a rational fraction \(\lambda\) such that \(L(x^{n})=\lambda(n) x^{n-m}\) for a fixed \(m\) and all integers \(n\). Actually, by setting \(z=x^{m}\), this kind of equation is reduced to a (generalized) hypergeometric equation for which this conjecture is known to be true [\textit{F. Beukers} and \textit{G. Heckman}, Invent. Math. 95, 325-354 (1989; Zbl 0663.30044)]. The more attractive part of the paper is concerned with the study of bad prime numbers. Namely, given an hypergeometric equation with finite Galois group, a prime number is said to be bad if either the equation cannot be reduced modulo \(p\) or the reduced equation does not have a fundamental system of rational solutions. In the case of the ordinary hypergeometric equation (satisfied by \(F_{1,2}\)) bad primes are shown to be exactly those for which the equation cannot be reduced. This is no longer true in the generalized case. Interesting examples are given.