Monodromie unipotente maximale, congruences “à la Lucas” et indépendance algébrique

Abstract

Let f ( z ) f(z) be in 1 + z Q [ [ z ] ] 1+z\mathbb {Q}[[z]] and S \mathcal {S} be an infinite set of prime numbers such that, for all p ∈ S p\in \mathcal {S} , we can reduce f ( z ) f(z) modulo p p . We let f ( z ) ∣ p f(z)_{\mid p} denote the reduction of f ( z ) f(z) modulo p p . Generally, when f ( z ) f(z) is D-finite, f ∣ p ( z ) f_{\mid p}(z) is algebraic over F p ( z ) \mathbb {F}_p(z) . It turns out that if f ∣ p ( z ) f_{\mid p}(z) is a solution of a polynomial of the form X − A p ( z ) X p l X-A_p(z)X^{p^l} , we can use this type of equations to obtain results of transcendence and algebraic independence over Q ( z ) \mathbb {Q}(z) . In the present paper, we look for conditions on the differential operators annihilating f ( z ) f(z) to guarantee the existence of these particular equations. Suppose that f ( z ) f(z) is solution of a differential operator H ∈ Q ( z ) [ d / d z ] \mathcal {H}\in \mathbb {Q}(z)[d/dz] having a strong Frobenius structure for all p ∈ S p\in \mathcal {S} and we also suppose that f ( z ) f(z) annihilates a Fuchsian differential operator D ∈ Q ( z ) [ d / d z ] \mathcal {D}\in \mathbb {Q}(z)[d/dz] such that zero is a regular singular point of D \mathcal {D} and the exponents of D \mathcal {D} at zero are equal to zero. Our main result states that, for almost every prime p ∈ S p\in \mathcal {S} , f ∣ p ( z ) f_{\mid p}(z) is solution of a polynomial of the form X − A p ( z ) X p l X-A_p(z)X^{p^l} , where A p ( z ) A_p(z) is a rational function with coefficients in F p \mathbb {F}_p of height less than or equal to C p 2 l Cp^{2l} with C C a positive constant that does not depend on p p . We also study the algebraic independence of these power series over Q ( z ) \mathbb {Q}(z) .