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Ranks of twists of elliptic curves and Hilbert’s tenth problem, arxiv:0904.3709v2 [math.NT
"... Abstract. In this paper we investigate the 2Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2Selmer rank, and we give lower bounds for the number of twists (with bound ..."
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Abstract. In this paper we investigate the 2Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2Selmer rank. As a consequence, under appropriate hypotheses we can find many twists with trivial MordellWeil group, and (assuming the ShafarevichTate conjecture) many others with infinite cyclic MordellWeil group. Using work of Poonen and Shlapentokh, it follows from our results that if the ShafarevichTate conjecture holds, then Hilbert’s Tenth Problem has a negative answer over the ring of integers of every number field. 1. Introduction and
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 18 (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 .
Hilbert’s Tenth Problem for algebraic function fields of characteristic 2
 Pacific J. Math
, 2003
"... Let K be an algebraic function field of characteristic 2 with constant field CK. Let C be the algebraic closure of a finite field in K. Assume that C has an extension of degree 2. Assume that there are elements u, x of K with u transcendental over CK and x algebraic over C(u) and such that K = CK(u, ..."
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Let K be an algebraic function field of characteristic 2 with constant field CK. Let C be the algebraic closure of a finite field in K. Assume that C has an extension of degree 2. Assume that there are elements u, x of K with u transcendental over CK and x algebraic over C(u) and such that K = CK(u, x). Then Hilbert’s Tenth Problem over K is undecidable. Together with Shlapentokh’s result for odd characteristic this implies that Hilbert’s Tenth Problem for any such field K of finite characteristic is undecidable. In particular, Hilbert’s Tenth Problem for any algebraic function field with finite constant field is undecidable. 1. Introduction. Hilbert’s Tenth Problem in its original form can be stated in the following form: Is there a uniform algorithm that determines, given a polynomial equation with integer coefficients, whether the equation has an integer solution
On Diophantine definability and decidability in large subrings of totally real number fields and their totally complex extensions of degree 2
, 2001
"... Let M be a number field. Let W be a set of nonarchimedean primes of M . Let OM,W = {x # M  ord p x # 0 #p ## W}. The author continues her investigation of Diophantine definability and decidability in rings OM,W where W is infinite. In this paper she improves her previous density estimat ..."
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Cited by 13 (11 self)
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Let M be a number field. Let W be a set of nonarchimedean primes of M . Let OM,W = {x # M  ord p x # 0 #p ## W}. The author continues her investigation of Diophantine definability and decidability in rings OM,W where W is infinite. In this paper she improves her previous density estimates and extends the results to the totally complex extensions of degree 2 of the totally real fields. In particular, the following results are proved. Let M be a totally real field or a totally complex extension of degree 2 of a totally real field. Then for any # > 0 there exists a set WM of primes of M whose density is bigger than 1  [M : Q] 1  # and such that Z has a Diophantine definition over OM,WM . (Thus, Hilbert's Tenth Problem is undecidable in OM,WM .) Let M be as above and let # > 0 be given. Let SQ be the set of all rational primes splitting in M . (If the extension is Galois but not cyclic, SQ contains all the rational primes.) Then there exists a set of Mprimes WM such that the set of rational primes WQ below WM di#ers from SQ by a set contained in a set of density less than # and such that Z has a Diophantine definition over OM,WM . (Again this will imply that Hilbert's Tenth Problem is undecidable in OM,WM .) 1
Hilbert’s tenth problem for algebraic function fields over infinite fields of constants of positive characteristic
 Pacific Journal of Mathematics
, 2000
"... Let K be an algebraic function field of characteristic p>2. Let C be the algebraic closure of a finite field in K. Assume that C has an extension of degree p. Assume also that K contains a subfield K1, possibly equal to C, and elements u, x such that u is transcendental over K1, x is algebraic ov ..."
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Let K be an algebraic function field of characteristic p>2. Let C be the algebraic closure of a finite field in K. Assume that C has an extension of degree p. Assume also that K contains a subfield K1, possibly equal to C, and elements u, x such that u is transcendental over K1, x is algebraic over C(u) and K = K1(u, x). Then the Diophantine problem of K is undecidable. Let G be an algebraic function field in one variable whose constant field is algebraic over a finite field and is not algebraically closed. Then for any prime p of G, the set of elements of G integral at p is Diophantine over G. 1. Introduction. The interest in the questions of Diophantine definability and decidability goes back to a question which was posed by Hilbert: Given an arbitrary polynomial equation in several variables over Z, is there a uniform algorithm
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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The purpose of this paper is to explore a conjecture due to Barry Mazur and formulated
Extensions of Büchi’s problem : Questions of decidability for addition and kth powers
"... The authors thank the referee for his comments. The second author aknowledges the hospitality of the University of CreteHeraklion, where the main part of this work was done. Abstract. We generalize a question of Büchi: Let R be an integral domain, C a subring and k ≥ 2 an integer. Is there an algo ..."
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Cited by 9 (5 self)
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The authors thank the referee for his comments. The second author aknowledges the hospitality of the University of CreteHeraklion, where the main part of this work was done. Abstract. We generalize a question of Büchi: Let R be an integral domain, C a subring and k ≥ 2 an integer. Is there an algorithm to decide the solvability in R of any given system of polynomial equations, each of which is linear in the k−th powers of the unknowns, with coefficients in C? We state a numbertheoretical problem, depending on k, a positive answer to which would imply a negative answer to the question for R = C = Z. We reduce a negative answer for k = 2 and for R = F (t), a field of rational functions of zero characteristic, to the undecidability of the ring theory of F (t). We address the similar question, where we allow, along with the equations, also conditions of the form ‘x is a constant ’ and ‘x takes the value 0 at t = 0’, for k = 3 and for function fields R = F (t) of zero characteristic, with C = Z[t]. We prove that a negative answer to this question would follow from a negative answer for a ring between Z and the extension of Z by a primitive cube root of 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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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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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.