  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.
