I'm a little new to this stuff so I might not know exactly what you're asking.

I believe there is a conjecture of Mazur that implies the type of description of $\Bbb Z$ you are looking for is impossible.

[This paper][1] by Poonen would be a good place to start. Of course Poonen would be able to give a much more satisfying answer. I caught the end of his talk about this stuff at the Joint Meetings.


  [1]: http://www-math.mit.edu/~poonen/papers/subrings.pdf