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15
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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Cited by 23 (3 self)
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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 12 (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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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 (7 self)
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The purpose of this paper is to explore a conjecture due to Barry Mazur and formulated
Diophantine definability of infinite discrete nonarchimedean sets and Diophantine models for large subrings of number fields
"... We prove that some infinite padically discrete sets have Diophantine definitions in large subrings of number fields. First, if K is a totally real number field or a totally complex degree2 extension of a totally real number field, then for every prime p of K there exists a set of Kprimes S of d ..."
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Cited by 7 (6 self)
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We prove that some infinite padically discrete sets have Diophantine definitions in large subrings of number fields. First, if K is a totally real number field or a totally complex degree2 extension of a totally real number field, then for every prime p of K there exists 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.
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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Cited by 5 (5 self)
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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.
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 4 (4 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 Undecidability of Function Fields of Characteristic Greater Than 2, Finitely Generated over Fields Algebraic over a Finite Field
, 2001
"... Let F be a function field of characteristic p>2, finitely generated over a field C algebraic over a finite field Cp and such that it has an extension of degree p. Then Hilbert's Tenth Problem is not decidable over F . 1 ..."
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Cited by 4 (3 self)
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Let F be a function field of characteristic p>2, finitely generated over a field C algebraic over a finite field Cp and such that it has an extension of degree p. Then Hilbert's Tenth Problem is not decidable over F . 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 4 (3 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.