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Common divisors of elliptic divisibility sequences over function fields
, 2004
"... Abstract. Let E/k(T) be an elliptic curve defined over a rational function field of characteristic zero. Fix a Weierstrass equation for E. For points R ∈ E(k(T)), write xR = AR /D2 R with relatively prime polynomials AR(T), DR(T) ∈ k[T]. The sequence {DnR} n≥1 is called the elliptic divisibility se ..."
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Abstract. Let E/k(T) be an elliptic curve defined over a rational function field of characteristic zero. Fix a Weierstrass equation for E. For points R ∈ E(k(T)), write xR = AR /D2 R with relatively prime polynomials AR(T), DR(T) ∈ k[T]. The sequence {DnR} n≥1 is called the elliptic divisibility sequence of R. Let P, Q ∈ E(k(T)) be independent points. We conjecture that and that deg ( gcd(DnP,DmQ) ) is bounded for m, n ≥ 1, gcd(DnP,DnQ) = gcd(DP,DQ) for infinitely many n ≥ 1. We prove these conjectures in the case that j(E) ∈ k. More generally, we prove analogous statements with k(T) replaced by the function field of any curve and with P and Q allowed to lie on different elliptic curves. If instead k is a finite field of characteristic p and again assuming that j(E) ∈ k, we show that deg ( gcd(DnP,DnQ)) is as large as n + O ( √ n) for infinitely many n ̸ ≡ 0 (mod p).
The sign of an elliptic divisibility sequence
, 2004
"... Abstract. An elliptic divisibility sequence (EDS) is a sequence of integers (Wn)n≥0 generated by the nonlinear recursion satisfied by the division polyomials of an elliptic curve. We give a formula for the sign of Wn for unbounded nonsingular EDS, a typical case being Sign(Wn) = (−1) ⌊nβ ⌋ for an i ..."
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Cited by 6 (0 self)
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Abstract. An elliptic divisibility sequence (EDS) is a sequence of integers (Wn)n≥0 generated by the nonlinear recursion satisfied by the division polyomials of an elliptic curve. We give a formula for the sign of Wn for unbounded nonsingular EDS, a typical case being Sign(Wn) = (−1) ⌊nβ ⌋ for an irrational number β ∈ R. As an application, we show that the associated sequence of absolute values (Wn) cannot be realized as the fixed point counting sequence of any abstract dynamical system.
Strong Divisibility Sequences and Some Conjectures.” The Fibonacci Quarterly 17
, 1979
"... Which recurrent sequences {tn: n = 0, 1,...} satisfy the following equation for greatest common divisors: (1) (tm,tn) = t(m>n) for all 772, n> 1, or the weaker divisibility property: ..."
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Cited by 5 (0 self)
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Which recurrent sequences {tn: n = 0, 1,...} satisfy the following equation for greatest common divisors: (1) (tm,tn) = t(m>n) for all 772, n> 1, or the weaker divisibility property:
Generalized greatest common divisors, divisibility sequences, and Vojta’s conjecture for blowups, Monatsh
 Math
"... Abstract. We apply Vojta’s conjecture to blowups and deduce a number of deep statements regarding (generalized) greatest common divisors on varieties, in particular on projective space and on abelian varieties. Special cases of these statements generalize earlier results and conjectures. We also dis ..."
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Cited by 3 (0 self)
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Abstract. We apply Vojta’s conjecture to blowups and deduce a number of deep statements regarding (generalized) greatest common divisors on varieties, in particular on projective space and on abelian varieties. Special cases of these statements generalize earlier results and conjectures. We also discuss the relationship between generalized greatest common divisors and the divisibility sequences attached to algebraic groups, and we apply Vojta’s conjecture to obtain a strong bound on the divisibility sequences attached to abelian varieties of dimension at least two.
Generating Functions of Linear Divisibility Sequences." The Fibonacci Quarterly 18
, 1987
"... A fcthorder divisibility sequence is introduced in Hall [3] as a sequence of rational integers uQ, uls u2,...9un9... satisfying a linear recurrence relation (1) un + k = a1un+k_1 + • • • • + akun9 ..."
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Cited by 2 (1 self)
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A fcthorder divisibility sequence is introduced in Hall [3] as a sequence of rational integers uQ, uls u2,...9un9... satisfying a linear recurrence relation (1) un + k = a1un+k_1 + • • • • + akun9
1 On The Elliptic Divisibility Sequences over Finite
"... Abstract—In this work we study elliptic divisibility sequences over finite fields. Morgan Ward in [11, 12] gave arithmetic theory of elliptic divisibility sequences. We study elliptic divisibility sequences, equivalence of these sequences and singular elliptic divisibility sequences over finite fiel ..."
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Abstract—In this work we study elliptic divisibility sequences over finite fields. Morgan Ward in [11, 12] gave arithmetic theory of elliptic divisibility sequences. We study elliptic divisibility sequences, equivalence of these sequences and singular elliptic divisibility sequences over finite fields Fp, p>3 is a prime. Keywords—Elliptic divisibility sequences, equivalent sequences, singular sequences. I. PRELIMINARIES. A divisibility sequence is a sequence (hn) (n ∈ N) of positive integers with the property that hmhn if mn. The oldest example of a divisibility sequence is the Fibonacci sequence. There are also divisibility sequences satisfying a nonlinear recurrence relation. These are the elliptic divisibility sequences and this relation comes from the recursion formula for elliptic division polynomials associated to an elliptic curve. An elliptic divisibility sequence (or EDS) is a sequence of integers (hn) satisfying a nonlinear recurrence relation hm+nhm−n = hm+1hm−1h 2 n − hn+1hn−1h 2 m (1) and with the divisibility property that hm divides hn whenever m divides n for all m ≥ n ≥ 1. There are some trivial examples such as the sequence of integers Z 0, 1, 2, 3, 4, 5, 6, ··· is an EDS but nontrivial examples abound. The simplest EDS is the sequence