Divisibility sequences for elliptic curves with complex multiplication

Streng, Marco

doi:10.2140/ant.2008.2.183

Cited by 23 publications

(30 citation statements)

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“…The properties of EDS were first studied formally by Ward [29]. The author [19] gave an ineffective proof of the existence of primitive divisors in EDS, and there are many papers giving quantitative and/or effective estimates; see for example [11,13,24,28]. Poonen [18] used EDS primitive divisors to resolve certain cases of Hilbert's 10 th problem in number fields.…”

Section: A Brief History Of Primitive Prime Divisorsmentioning

confidence: 99%

Primitive divisors, dynamical Zsigmondy sets, and Vojtaʼs conjecture

Silverman¹

2013

Journal of Number Theory

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A primitive prime divisor of an element a n of a sequence (a m ) m≥0 is a prime p that divides a n , but does not divide a m for all m < n. The Zsigmondy set Z of the sequence is the set of n such that a n has no primitive prime divisors. Let f : X → X be a self-morphism of a variety, let D be an effective divisor on X, and let P ∈ X, all defined overQ. We consider the Zsigmondy set Z(X, f, P, D) of the sequence defined by the arithmetic intersection of the f -orbit of P with D. Under various assumptions on X, f , D, and P , we use Vojta's conjecture with truncated counting function to prove that the set of points f n (P ) with n ∈ Z(X, f, P, D) is not Zariski dense in X. The classical definition does not include the requirement that a m = 0, but including this condition allows for cleaner statements of theorems. Note that if some a m = 0, then a m is divisible by every prime, so if we didn't exclude such a m , then a n with n > m would never have a primitive prime divisor.

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Section: A Brief History Of Primitive Prime Divisorsmentioning

confidence: 99%

Primitive divisors, dynamical Zsigmondy sets, and Vojtaʼs conjecture

Silverman¹

2013

Journal of Number Theory

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“…Elliptic divisibility sequences were first studied formally by Ward [53,54], although Watson [55] considered related sequences in his resolution of Lucas' square pyramid problem. Recently, the study of elliptic divisibility sequences has seen renewed interest [15,41,43,44,46,49], including applications to Hilbert's 10th problem [6,11,29] and cryptography [24,36,45]. (We remark that some authors use a slightly different definition of EDS via the division polynomial recursion.…”

Section: History and Motivationmentioning

confidence: 99%

Algebraic Divisibility Sequences Over Function Fields

Ingram

Mahé

Silverman

et al. 2012

J. Aust. Math. Soc.

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In this note we study the existence of primes and of primitive divisors in function field analogues of classical divisibility sequences. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.2010 Mathematics subject classification: primary 11B39; secondary 11G05. Keywords and phrases: lucas sequence, elliptic divisibility sequence, primitive divisor, function field over number field.

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“…The previous theorem was generalized by Streng in [11] to the case when O is not Z. Theorem 1.4 [11, Principal case of the main theorem, Streng, 2008] Let E be an elliptic curve defined over a number field K by a Weierstrass equation with coefficients in O K . Take P in E(K ).…”

Section: Introductionmentioning

confidence: 99%

Primitive divisors of sequences associated to elliptic curves with complex multiplication

Verzobio

2021

Res. number theory

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Let P and Q be two points on an elliptic curve defined over a number field K. For $$\alpha \in {\text {End}}(E)$$ α ∈ End ( E ) , define $$B_\alpha $$ B α to be the $$\mathcal {O}_K$$ O K -integral ideal generated by the denominator of $$x(\alpha (P)+Q)$$ x ( α ( P ) + Q ) . Let $$\mathcal {O}$$ O be a subring of $${\text {End}}(E)$$ End ( E ) , that is a Dedekind domain. We will study the sequence $$\{B_\alpha \}_{\alpha \in \mathcal {O}}$$ { B α } α ∈ O . We will show that, for all but finitely many $$\alpha \in \mathcal {O}$$ α ∈ O , the ideal $$B_\alpha $$ B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms $$g\ne 0$$ g ≠ 0 and f so that $$f(P)= g(Q)$$ f ( P ) = g ( Q ) . This is a generalization of previous results on elliptic divisibility sequences.

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Divisibility sequences for elliptic curves with complex multiplication

Cited by 23 publications

References 16 publications

Primitive divisors, dynamical Zsigmondy sets, and Vojtaʼs conjecture

Primitive divisors, dynamical Zsigmondy sets, and Vojtaʼs conjecture

Algebraic Divisibility Sequences Over Function Fields

Primitive divisors of sequences associated to elliptic curves with complex multiplication

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