Book contents
- Frontmatter
- Contents
- Acknowledgements
- 1 Introduction
- 2 Diophantine classes: definitions and basic facts
- 3 Diophantine equivalence and Diophantine decidability
- 4 Integrality at finitely many primes and divisibility of order at infinitely many primes
- 5 Bound equations for number fields and their consequences
- 6 Units of rings of W-integers of norm 1
- 7 Diophantine classes over number fields
- 8 Diophantine undecidability of function fields
- 9 Bounds for function fields
- 10 Diophantine classes over function fields
- 11 Mazur's conjectures and their consequences
- 12 Results of Poonen
- 13 Beyond global fields
- Appendix A Recursion (computability) theory
- Appendix B Number theory
- References
- Index
11 - Mazur's conjectures and their consequences
Published online by Cambridge University Press: 14 October 2009
- Frontmatter
- Contents
- Acknowledgements
- 1 Introduction
- 2 Diophantine classes: definitions and basic facts
- 3 Diophantine equivalence and Diophantine decidability
- 4 Integrality at finitely many primes and divisibility of order at infinitely many primes
- 5 Bound equations for number fields and their consequences
- 6 Units of rings of W-integers of norm 1
- 7 Diophantine classes over number fields
- 8 Diophantine undecidability of function fields
- 9 Bounds for function fields
- 10 Diophantine classes over function fields
- 11 Mazur's conjectures and their consequences
- 12 Results of Poonen
- 13 Beyond global fields
- Appendix A Recursion (computability) theory
- Appendix B Number theory
- References
- Index
Summary
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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- Information
- Hilbert's Tenth ProblemDiophantine Classes and Extensions to Global Fields, pp. 180 - 188Publisher: Cambridge University PressPrint publication year: 2006