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Could you suggest me a book or a link where I can find some information about the Grothendieck problem about differential equations?

The Grothendieck problem that I am reffering to is the following:

$$\alpha_n(x)y^{(n)}(x)+\dots +a_1 (x)y'(x)+a_0(x)y(x)=0, a_i \in \mathbb{Z}[x]\ \ \ \ (*)$$

We suppose that for almost each prime $p$, $(*)$, modulo $p$, has $n$ linearly independent solutions (powerseries in $\mathbb{F}_p((x))$, with field of constants $\mathbb{F}_p((x^p))$). Then $(*)$ has $n$ linearly independent solutions (powerseries in $\mathbb{C}((x))$ with field of constants $\mathbb{C}((x))$) and all are algebraic.

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    $\begingroup$ Are you talking about the Grothendieck-Katz conjecture? Here is the link to the Wikipedia page: en.wikipedia.org/wiki/…. $\endgroup$
    – Jason Starr
    Commented Oct 1, 2015 at 9:56
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    $\begingroup$ I cannot say immediately that your formulation is the same as the standard formulation of the Grothendieck-Katz conjecture, but they look quite similar. $\endgroup$
    – Jason Starr
    Commented Oct 1, 2015 at 10:10
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    $\begingroup$ @JasonStarr Looks identical, actually... $\endgroup$
    – Igor Rivin
    Commented Oct 1, 2015 at 15:31
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    $\begingroup$ The information on the wikipedia link is quite good. There are a few other references that I would add, notably Andre's book "G-functions and Geometry" and a couple of papers by D. and G. Chudnovsky, but they can all be found in the bibliography of, say, Chambert-Loir's paper linked to there. The list is then pretty much exhaustive, and it is up to you what to get out of it. (What kind of information do you look for? Do you for instance have the algebro-geometric background to follow Katz's Inventiones paper that contains the proof of the conjecture in the case of a Picard-Fuchs equation?) $\endgroup$
    – Vesselin Dimitrov
    Commented Oct 8, 2015 at 18:46
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    $\begingroup$ D. and G. Chudnovsky have solved other cases by elementary methods of diophantine approximations (Pade approximants). Chambert-Loir's paper is probably the best place to start on this; then you may look into the original papers by the Chudnovskys. $\endgroup$
    – Vesselin Dimitrov
    Commented Oct 8, 2015 at 18:48

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