There is an outrageous conjecture which I abbreviate to:

Algebraic Integrability Conjecture: on a complex variety $V$, a foliation $\mathcal{F}\subset T_{V/\mathbb{C}}$ is given by some rational fibration $f:V \dashrightarrow W$, i.e., $\mathcal{F}=T_f$, (algebraically integrable) if and only if the reduction modulo $p$ of the foliation is $p$-closed for almost all primes.

Being $p$-closed means that if $D\in \mathcal{F}$, then its $p$-times composition, which is also a derivation (!) if $p=0$, is also there, i.e., $D^p\in \mathcal{F}$. It is a positive characteristic feature.

This conjecture is outrageous because it mixes complex analysis with the modulo $p$, and it has strong implications for monodromy theory: it would characterize the systems with finite ones (something topological, analytical) by their arithmetic properties.

Some history. There is something called the Poincaré--Painlevé problem (1890s): for a vector field with rational coefficients on a plane $D=f\partial_x +g\partial_y$, where $f,g\in \mathbb{C}(x,y)$. It is hard to compute explicit solutions to equations like $D(F)=0$; thus, Poincaré asked whether we could bound the genus of any $D$-invariant curve in terms of the degrees of $ f$ and $ g$: we cannot. One is required to add more global invariants to do anything about it.

From a different angle, Grothendieck (1960s), while working on crystalline cohomology, suggested that bundles with a connection which, modulo almost all primes, is $p$-flat, should be algebraic fibrations. We know some cases by Katz, Bost, and others. Katz was the first to write about it explicitly and prove the first cases; thus, we call it the Grothendieck--Katz conjecture.

However, bundles with $p$-flat connections are just a special case of foliations that are $p$-closed, thus in a never-published and not anymore publicly available preprint, Ekedahl, Shephard-Baron, and Taylor (1990s) proposed ``Conjecture F'', which is what I call the algebraic integrability conjecture, because calling it:

Poincaré--Painlevé--Grothendieck--Katz--Ekedahl--Shephard-Baron--Taylor Conjecture/Problem

is ridiculous. Moreover, some people like to joke about the authors' opinions, whether the conjecture is true or not, since each supposedly had a different one. True, false, ?. This is just a repeat of jokes; I do not claim that this is factual.

Furthermore, as a recent survey Algebraic solutions of linear differential equations: an arithmetic approach explains, this conjecture would also tell us a lot about hypergeometric series, and any other series, about when they are rational functions and when not, simply by computing some easy modulo $p$ data.

Naively, one might think of just computing the solution and checking the equations to verify its existence: both conditions are kinds of equations to be satisfied, there is inclusion, but why these big sets of equations are equal, who knows? There are infinitely many of them, and even finitely many of them grow too fast with bigger degrees of coefficients and solutions we allow...

Finally, this is a conjecture about geometric objects, but I cannot imagine how something so general could be proved geometrically, because the number of invariants and their relations grow rapidly with dimension and vary with rank, the variety it is on. Therefore, a proof would have to give us a way to go through all the geometries of varieties and, at the same time, ignore them all...