Alg\'ebricit\'e modulo p, s\'eries hyperg\'eom\'etriques et structures de Frobenius forte

TL;DR

This paper explores the algebraicity of Siegel's G-functions modulo p, emphasizing the role of strong Frobenius structures, and applies these insights to a conjecture on the algebraic degree of G-functions.

Contribution

It explicitly relates strong Frobenius structures to algebraicity degrees of G-functions modulo p and proves the conjecture for hypergeometric series using these structures.

Findings

Reduction of G-functions modulo p is algebraic with degree bounded by p^{n^2h}.

Fuchsian operators with rational exponents and rigid monodromy have strong Frobenius structures for almost all primes.

The results confirm the Adamczewski-Delaygue conjecture for generalized hypergeometric series.

Abstract

This work is devoted to study of algebraicty modulo p of Siegel's G-functions. Our goal is to emphasize the relevance of the notion of strong Frobenius structure, clasically studied in the theory of the p-adic diffenrential equations, for the study of a Adamczewski-Delaygue's conjecture concerning of the degree of algebraicity modulo p of G-functions. For this, we first make a Christol's result explicit by showing that if $f$ is a G-function that is solution of a differential operator $L$ in $ \mathbb{Q}(z)[d/dz]$ of order $n$ endowed of a strong Frobenius structure of period $h$ for the prime number $p$ and that $f(z)$ belongs to $\mathbb{Z}_{(p)}[[z]]$, then the reduction of $f$ modulo $ p $ is algebraic over $\mathbb F_p(z)$ and its algebraicity degree is bounded by $p^{n^2h}$. By generalizing an approach introduced by Salinier, we show that if $L$ is a Fuchsian operator with coefficients in $\mathbb{Q}(z)$, whose monodromy group is rigid and whose exponents are rational numbers, then $L$ has for almost every prime number $p$ a strong Frobenius structure of period $h$, where $h$ is explicitly bounded and does not depend on $p$. A slightly different version of this result has been recently demonstrated by Crew following a different approach based on the $p$ -adic cohomology. We use these two results to solve the mentioned conjecture in the case of generalized hypergeometric series.