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Hilbert’s tenth problem and Mazur’s conjecture for large subrings of Q
 J. Amer. Math. Soc
"... Abstract. We give the first examples of infinite sets of primes S such that Hilbert’s Tenth Problem over Z[S −1] has a negative answer. In fact, we can take S to be a density 1 set of primes. We show also that for some such S there is a punctured elliptic curve E ′ over Z[S −1] such that the topolog ..."
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Abstract. We give the first examples of infinite sets of primes S such that Hilbert’s Tenth Problem over Z[S −1] has a negative answer. In fact, we can take S to be a density 1 set of primes. We show also that for some such S there is a punctured elliptic curve E ′ over Z[S −1] such that the topological closure of E ′ (Z[S −1]) in E ′ (R) has infinitely many connected components. 1.
Using Elliptic Curves of Rank One towards the Undecidability of Hilbert's Tenth Problem over Rings of Algebraic Integers
"... Let F be their rings of integers. If there exists an elliptic curve E over F such that rk E(F ) = rk E(K) = 1, then there exists a diophantine definition of . ..."
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Cited by 19 (2 self)
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Let F be their rings of integers. If there exists an elliptic curve E over F such that rk E(F ) = rk E(K) = 1, then there exists a diophantine definition of .
DEFINING INTEGRALITY AT PRIME SETS OF HIGH DENSITY IN NUMBER FIELDS
 VOL. 101, NO. 1 DUKE MATHEMATICAL JOURNAL
, 2000
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DIOPHANTINE DEFINABILITY OF INFINITE DISCRETE NONARCHIMEDEAN SETS AND DIOPHANTINE MODELS OVER Large Subrings Of Number Fields
, 2004
"... We prove that infinite padically discrete sets have Diophantine definitions in large subrings of some number fields. First, if K is a totally real number field or a totally complex degree2 extension of a totally real number field, then there exists a prime p of K and a set of Kprimes S of densit ..."
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Cited by 11 (8 self)
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We prove that infinite padically discrete sets have Diophantine definitions in large subrings of some number fields. First, if K is a totally real number field or a totally complex degree2 extension of a totally real number field, then there exists a prime p of K and a set of Kprimes S of density arbitrarily close to 1 such that there is an infinite padically discrete set that is Diophantine over the ring OK,S of Sintegers in K. Second, if K is a number field over which there exists an elliptic curve of rank 1, then there exists a set of Kprimes S of density 1 and an infinite Diophantine subset of OK,S that is vadically discrete for every place v of K. Third, if K is a number field over which there exists an elliptic curve of rank 1, then there exists a set of Kprimes S of density 1 such that there exists a Diophantine model of Z over OK,S. This line of research is motivated by a question of Mazur concerning the distribution of rational points on varieties in a nonarchimedean topology and questions concerning extensions of Hilbert’s Tenth Problem to subrings of number fields.
A ring version of Mazur’s conjecture on topology of rational points
 Internat. Math. Res. Notices
"... The purpose of this paper is to explore a conjecture due to Barry Mazur and formulated ..."
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Cited by 10 (6 self)
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The purpose of this paper is to explore a conjecture due to Barry Mazur and formulated
ELLIPTIC CURVES RETAINING THEIR RANK IN FINITE EXTENSIONS AND HILBERT’S TENTH PROBLEM FOR RINGS OF ALGEBRAIC NUMBERS
, 2008
"... Abstract. Using Poonen’s version of the “weak vertical method ” we produce new examples of “large ” and “small ” rings of algebraic numbers (including rings of integers) where Z and/or the ring of integers of a subfield are existentially definable and/or where the ring version of Mazur’s conjecture ..."
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Cited by 6 (6 self)
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Abstract. Using Poonen’s version of the “weak vertical method ” we produce new examples of “large ” and “small ” rings of algebraic numbers (including rings of integers) where Z and/or the ring of integers of a subfield are existentially definable and/or where the ring version of Mazur’s conjecture on the topology of rational points does not hold. 1.
DIOPHANTINE DEFINABILITY AND DECIDABILITY IN THE EXTENSIONS OF DEGREE 2 OF TOTALLY REAL FIELDS
, 2006
"... Abstract. We investigate Diophantine definability and decidability over some subrings of algebraic numbers contained in quadratic extensions of totally real algebraic extensions of Q. Among other results we prove the following. The big subring definability and undecidability results previously shown ..."
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Cited by 6 (4 self)
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Abstract. We investigate Diophantine definability and decidability over some subrings of algebraic numbers contained in quadratic extensions of totally real algebraic extensions of Q. Among other results we prove the following. The big subring definability and undecidability results previously shown by the author to hold over totally complex extensions of degree 2 of totally real number fields, are shown to hold for all extensions of degree 2 of totally real number fields. The definability and undecidability results for integral closures of “small ” and “big ” subrings of number fields in the infinite algebraic extensions of Q, previously shown by the author to hold for totally real fields, are extended to a large class of extensions of degree 2 of totally real fields. This class includes infinite cyclotomics and abelian extensions with finitely many ramified rational primes. 1.
On diophantine definability and decidability in some infinite totally real extensions
 of Q, Trans. Amer. Math. Soc
"... Abstract. Let M be a number field, and WM a set of its nonArchimedean primes. Then let OM,W = {x ∈ M  ordt x ≥ 0, ∀t � ∈ WM}. Let{p1,...,pr} M be a finite set of prime numbers. Let Finf be the field generated by all the p j ith roots of unity as j → ∞ and i = 1,...,r. Let Kinf be the largest tot ..."
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Abstract. Let M be a number field, and WM a set of its nonArchimedean primes. Then let OM,W = {x ∈ M  ordt x ≥ 0, ∀t � ∈ WM}. Let{p1,...,pr} M be a finite set of prime numbers. Let Finf be the field generated by all the p j ith roots of unity as j → ∞ and i = 1,...,r. Let Kinf be the largest totally real subfield of Finf. Then for any ε>0, there exist a number field M ⊂ Kinf,andasetWMof nonArchimedean primes of M such that WM has density greater than 1 − ε, andZhas a Diophantine definition over the integral closure of OM,W in Kinf. M 1.
Hilbert’s Tenth Problem over Rings of NumberTheoretic Interest
"... 2. The original problem 1 3. Turing machines and decision problems 2 4. Recursive and listable sets 3 ..."
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2. The original problem 1 3. Turing machines and decision problems 2 4. Recursive and listable sets 3