What does a solution of a Diophantine equation in $\mathbb{F}_{p}$ tell about general integer solution?

Question

Q: What does a solution of a Diophantine equation in $\mathbb{F}_{p^k}$ for all $k\in\mathbb{N}$ and prime $p$ tell about general integer solution?

We know that if there are not any solutions of a Diophantine equation in $\mathbb{F}_p$ then there are not any solutions in integers. Is this the only reason to study solutions in $\mathbb{F}_p$ or there are other reasons?

I read these two Wikipedia articles 1 and 2 but couldn't find answer to this question.

Answer 1

For example, if $E$ is an elliptic curve over $\Bbb Q$, the number of points of $E$ over all finite fields defines the Hasse-Weil L-function of $E$. The conjectures of Birch and Swinnerton-Dyer indicate that this L-function determines the rank of the group of rational points on $E$ etc.

Answer 2

I'm not sure if this is what you are looking for, but it might be interesting for you.

If you have solutions for all $\mathbb{Z}/p\mathbb{Z}$, then sometimes under certain conditions we can lift these solutions for the higher modulo $p$ powers $\mathbb{Z}/p^{n}\mathbb{Z}$ (check [Hensel's Lemma][1]).

There is a ring, called the ring of $p$-adic integers, and denoted by $\mathbb{Z}_{p}$, which contains $\mathbb{Z}$ as a subring and is somehow "made up" by the finite quotients $\mathbb{Z}/p^{n}\mathbb{Z}$.

So for certain polynomial equations, this ring allows us to say things like: this equation has solutions in all the $\mathbb{Z}/p^{n}\mathbb{Z}$ if and only it has solutions in $\mathbb{Z}_{p}$, and so if this is the case, it is a matter of investigating whether or not this solution lands in the ring $\mathbb{Z}$.

I don't know if you can find such a thing for finite fields $\mathbb{F}_{p^{k}}$ (note that for $k>1$ these are not the quotient rings I was talking above).

I'm still a student far from expert; the main reasons I found in algebraic number theory so far for studying finite fields are:

1. When you study prime ideal decomposition in algebraic number fields (ramification, etc); for example, the decomposition $(p)=\mathfrak{P}_{1}\dots\mathfrak{P}_{k}$ of a rational prime $p$ in some $\mathcal{O}_{K}$ allows you to consider the finite field extensions $\mathcal{O}_{K}/\mathfrak{P}_{i}$ over $\mathbb{Z}/p\mathbb{Z}$; and this extension codifies some information about how $p$ decomposes.
2. In the local field case (which by the way has to do with the above $\mathbb{Z}_{p}$), there is a notion of Galois unramified extension; for such an extension, the Galois group is isomorphic to the Galois group of associated residue fields extension, which is a finite field extension; so, understanding Galois theory of finite fields helps you to get a handle on this important class of local field extensions.

I'm sure there are plenty of others I'm not aware about;