Yo, this is probably some nonsense stupid question, but I will ask it anyway.
Let $X$ be some variety over $\mathbb{Z}$. Suppose that I know $X (\mathbb{F}_{p^n})$ for every $n$. Does it tell me anything about $X (\mathbb{Q}_p)$?
The motivation is kind of idiot. A solution of a system of polynomial equations over $\mathbb{Q}_p$ is the same thing as a solution modulo $p^n$ for every $n$, so the collection of all $X (\mathbb{Z}/p^n)$ tell me exactly $X (\mathbb{Q}_p)$. So first order approximations of solutions over $\mathbb{Q}_p$ are exactly $X (\mathbb{F}_p)$.
More generally, what can be said for arbitrary global fields instead of simply $\mathbb{Q}$? I care mainly about number fields, but an answer about functional fields only would also be helpful.
Thanks in advance.
The issue here is that the behaviors of $\Bbb F_{p^n}$ and $\Bbb Z/p^n$ as $n\to\infty$ are two very different things. Taking $n\to\infty$ on the field side is done by taking a colimit, and gets you to $\overline{\Bbb F}_{p}$, the algebraic closure of $\Bbb F_p$. On the $\Bbb Z/p^n$ side, taking $n\to \infty$ happens via a limit, and the result is the $p$-adics.
For example, if one takes the variety given by $x^2+1=0$ where $p=3$, this variety won't have any points over $\Bbb Q_3$ or any ring of the form $\Bbb Z/3^n$, but it has points over $\Bbb F_{3^{2n}}$ for all $n$.
In general, determining the behavior of solutions over $\Bbb Q_p$ given the behavior over $\Bbb Z/p^n$ is a problem solved by Hensel's lemma, while there's generally not a relationship between the behavior of solutions over $\Bbb F_{p^n}$ and $\Bbb Q_p$. Hensel's lemma and concepts in $p$-adic analysis are where I think you might do well to poke around while developing the ideas contained in your question. If you're interested in comparing results modulo every prime and over $\Bbb Q$, you might want to look up the Hasse principle and counterexamples to it.