Poonen's theorem is arguably the most important development in the subject since Matiyasevich completed the proof of the original HTP in the late 1960s. One could say that for the first time the solution of HTP for Z has become visible over the distant horizon, though we still have to traverse an “infinite” distance, as will be explained below.

The result came out of the attempts to falsify the ring version of Mazur's conjecture (Conjecture 11.1.1) for a ring of rational S-integers where S has natural density equal to 1. In the process of constructing a counterexample to the conjecture, Poonen constructed a (tight) Diophantine model of Z over such a ring. This result has moved us “infinitely” far away from where we started (Z and rings of rational S-integers with finite S), but since the set of allowed denominators in Poonen's theorem still misses being an infinite set of primes (though of natural density zero), we still have “infinitely” far to go.

In this chapter we will go over Poonen's proof, which appeared originally in [74], in some detail. We will start with the overall plan and then will try to sort out the rather challenging technical details.

A statement of the main theorem and an overview of the proof

A good place to start is a precise statement of the theorem, which is presented below.