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Published online by Cambridge University Press: 14 October 2009
In this chapter we explore two conjectures due to Barry Mazur. These conjectures, which are a part of a series of conjectures made by Mazur concerning topology of rational points, have had a very important influence on the development of the subject. The conjectures first appeared in [55], and later in [56], [57], and [58]. They were explored further by among others, Colliot-Thélène, Skorobogatov, and Swinnerton-Dyer in [4], Cornelissen and Zahidi in [6], Pheidas in [70], and the present author in [108]. Perhaps the most spectacular result which has come out of attempts to prove or disprove the conjectures is a theorem of Poonen, which will be discussed in detail in the next chapter. Unfortunately, up to the time of writing, the conjectures are still unresolved.
The two conjectures
The first conjecture that we are going to discuss states the following.
Conjecture 11.1.1.Let V be any variety over Q. Then the topological closure of V(Q) in V(R) possesses at most a finite number of connected components. (Conjecture 2 of [58].)
Remark 11.1.2. Let W be an algebraic set defined over a number field.
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