Consider the Diophantine equation $f(x)=0$, where x is a vector of integers and $f: \mathbb Z^n \rightarrow \mathbb Z$ is a polynomial function. Is the following statement true?
The structure of the set of solutions to $f(x)=0$ is uniquely determined by the number of solutions to $f(x)\equiv 0 \pmod{p^i}$ for each prime $p$ and $i>0$.
If it is unknown whether the statement is true, please explain what we know about the statement.
In particular, do we know that this is true for elliptic curves or is it necessary to prove the Birch Swinnerton-Dyer conjecture to know this?