Grothendieck Problem: Solving $(*)$ Algebraically

[mathmari]

Hey!

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.

 

[mathmari]

(I think that this problem is called Grothendieck-Katz conjecture.)Could you explain to me the above problem when we have for example the differential equation $xy'-ky=0, k\in \mathbb{Z}$ ?

We have to find a prime such that for all primes $p$ greater or equal than that one, it stands that modulo $p$ the differential equation has in this case one solution in $\mathbb{F}_p((x))$.

Then the differential equation has one algebraic solution in $\mathbb{C}((x))$.

Is this correct?