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CHARACTERIZING INTEGERS AMONG RATIONAL NUMBERS WITH A UNIVERSALEXISTENTIAL FORMULA
"... Abstract. We prove that Z in definable in Q by a formula with two universal quantifiers followed by seven existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Qmorphisms, whether there exists one that is surjective on rational points. We also giv ..."
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Abstract. We prove that Z in definable in Q by a formula with two universal quantifiers followed by seven existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Qmorphisms, whether there exists one that is surjective on rational points. We also give a formula, again with universal quantifiers followed by existential quantifiers, that in any number field defines the ring of integers. 1. Introduction. 1.1. Background. D. Hilbert, in the 10th of his famous list of 23 problems, asked for an algorithm for deciding the solvability of any multivariable polynomial equation in integers. Thanks to the work of M. Davis, H. Putnam, J. Robinson [DPR61], and Y. Matijasevič [Mat70], we know that no such algorithm
THE UNIFORM PRIMALITY CONJECTURE FOR ELLIPTIC CURVES
"... Abstract. An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is shown to possess a uniformly bounded number of prime terms. This result applies over the rational numbers, assuming Lang’s conjecture, and over the rational function field, unconditionally. In th ..."
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Abstract. An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is shown to possess a uniformly bounded number of prime terms. This result applies over the rational numbers, assuming Lang’s conjecture, and over the rational function field, unconditionally. In the latter case, a uniform bound is obtained on the index of a prime term. Sharpened versions of these techniques are shown to lead to explicit results where all the irreducible terms can be computed. 1.
DESCENT ON ELLIPTIC CURVES AND HILBERT’S TENTH PROBLEM
"... Abstract. Descent via an isogeny on an elliptic curve is used to construct two subrings of the field of rational numbers, which are complementary in a strong sense, and for which Hilbert’s Tenth Problem is undecidable. This method further develops that of Poonen, who used elliptic divisibility seque ..."
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Abstract. Descent via an isogeny on an elliptic curve is used to construct two subrings of the field of rational numbers, which are complementary in a strong sense, and for which Hilbert’s Tenth Problem is undecidable. This method further develops that of Poonen, who used elliptic divisibility sequences to obtain undecidability results for some large subrings of the rational numbers. 1. Hilbert’s Tenth Problem In 1970, Matijasevič [10], building upon earlier work of Davis, Putnam and Robinson [5], resolved negatively Hilbert’s Tenth Problem for the ring Z, of rational integers. This means there is no general algorithm which will decide if a polynomial equation, in several variables, with integer coefficients has an integral solution. Equivalently, one says Hilbert’s Tenth Problem is undecidable for the integers. See [14, Chapter 1] for a full overview and background reading. The same problem, except now over the rational field Q, has not been resolved. In other words, it is not known if there is an algorithm which will decide if a polynomial equation with integer coefficients (or rational coefficients, it doesn’t matter) has a rational solution. Recently, Poonen [11] took a giant leap in this direction by proving the same negative result for some large subrings of Q. To make this precise, given a prime p of Z, let .p denote the usual padic absolute value. Let S denote a set of rational primes. Write ZS = Z[1/S] = {x ∈ Q: xp ≤ 1 for all p / ∈ S}, 1991 Mathematics Subject Classification. 11G05, 11U05. Key words and phrases. Elliptic curve, elliptic divisibility sequence, Hilbert’s
Defining the integers in large rings of a number field using one universal quantifier
, 2007
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USING INDICES OF POINTS ON AN ELLIPTIC CURVE TO CONSTRUCT A DIOPHANTINE MODEL OF Z AND DEFINE Z USING ONE UNIVERSAL QUANTIFIER IN VERY LARGE SUBRINGS OF NUMBER FIELDS, INCLUDING Q
, 2009
"... Let K be a number field and let E be an elliptic curve defined and of rank one over K. For a set WK of primes of K, let OK,WK = {x ∈ K: ordp x ≥ 0, ∀p ̸ ∈ WK}. Let P ∈ E(K) be a generator of E(K) modulo the torsion subgroup. Let (xn(P), yn(P)) be the affine coordinates of [n]P with respect to a fix ..."
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Let K be a number field and let E be an elliptic curve defined and of rank one over K. For a set WK of primes of K, let OK,WK = {x ∈ K: ordp x ≥ 0, ∀p ̸ ∈ WK}. Let P ∈ E(K) be a generator of E(K) modulo the torsion subgroup. Let (xn(P), yn(P)) be the affine coordinates of [n]P with respect to a fixed Weierstrass equation of E. We show that there exists a set WK of primes of K of natural density one such that in OK,WK multiplication of indices (with respect to some fixed multiple of P) is existentially definable and therefore these indices can be used to construct a Diophantine model of Z. We also show that Z is definable over OK,WK using just one universal quantifier. Both, the construction of a Diophantine model using the indices and the firstorder definition of Z can be lifted to the integral closure of OK,WK in any infinite extension K ∞ of K as long as E(K∞) is finitely generated and of rank one.
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"... Abstract. Elliptic divisibility sequences are integer recurrence sequences, each of which is associated to an elliptic curve over the rationals together with a rational point on that curve. In this paper we present a higherdimensional analogue over arbitrary base fields. Suppose E is an elliptic cu ..."
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Abstract. Elliptic divisibility sequences are integer recurrence sequences, each of which is associated to an elliptic curve over the rationals together with a rational point on that curve. In this paper we present a higherdimensional analogue over arbitrary base fields. Suppose E is an elliptic curve over a field K, and P1,..., Pn are points on E defined over K. To this information we associate an ndimensional array of values of K satisfying a complicated nonlinear recurrence relation. Arrays satisfying this relation are called elliptic nets. All elliptic nets arise from elliptic curves in this manner. In this paper we describe properties of elliptic nets, and in particular we prove an explicit bijection between the set
THE UNIFORM PRIMALITY CONJECTURE FOR THE TWISTED FERMAT CUBIC
"... Abstract. On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. We prove that the number of prime terms in the sequence is uniformly bounded. When the rational point is the image of another rational point un ..."
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Abstract. On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. We prove that the number of prime terms in the sequence is uniformly bounded. When the rational point is the image of another rational point under a certain 3isogeny, all terms beyond the first fail to be primes. 1.
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"... positive integers n dividing the nth term of an elliptic divisibility sequence ..."
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positive integers n dividing the nth term of an elliptic divisibility sequence
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"... positive integers n dividing the nth term of an elliptic divisibility sequence ..."
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positive integers n dividing the nth term of an elliptic divisibility sequence