Maximal Unipotent Monodromy, congruences "\`a la Lucas" and Algebraic independence

TL;DR

This paper investigates conditions under which solutions to certain differential equations exhibit algebraic independence and satisfy special congruences modulo primes, linking differential operators, Frobenius structures, and algebraic properties of power series.

Contribution

It establishes criteria for solutions of differential operators with Frobenius structures to satisfy Lucas-type congruences and explores their algebraic independence over ield.

Findings

Solutions satisfy polynomial equations of the form X-A_p(z)X^{p^l} for almost all primes p.

Provides bounds on the height of rational functions A_p(z) in these equations.

Shows algebraic independence of these power series over ield.

Abstract

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