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Title of Journal: manuscripta math

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Abbravation: manuscripta mathematica

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Springer-Verlag

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10.1016/0014-5793(80)81300-7

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1432-1785

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Common divisors of elliptic divisibility sequences

Authors: Joseph H Silverman
Publish Date: 2004/07/16
Volume: 114, Issue: 4, Pages: 431-446
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Abstract

Let E/kT be an elliptic curve defined over a rational function field of characteristic zero Fix a Weierstrass equation for E For points R ∈ EkT write xR=AR/DR2 with relatively prime polynomials A R TD R T ∈ kT The sequence D nR n ≥ 1 is called the elliptic divisibility sequence of R Let PQ ∈ EkT be independent points We conjecture that deg gcdD nP D mQ is bounded for m n ≥ 1 and that gcdD nP D nQ = gcdD P D Q for infinitely many n ≥ 1 We prove these conjectures in the case that jE ∈ k More generally we prove analogous statements with kT 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 jE ∈ k we show that deg gcdD nP D nQ is as large as Open image in new window for infinitely many n≢0 mod pAcknowledgements I would like to thank Gary Walsh for rekindling my interest in the arithmetic properties of divisibility sequences and for bringing to my attention the articles 1 and 3 and David McKinnon for showing me his article 14 I also want to thank Zeev Rudnick for his helpful comments concerning the first draft of this paper especially for Remark 5 for pointing out 7 and for letting me know that he described conjectures similar to those made in this paper at CNTA 7 in 2002


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  1. A stability criterion for nonparametric minimal submanifolds
  2. Harmonic maps and asymptotic Teichmüller space
  3. On the backward uniqueness property for the heat equation in two-dimensional conical domains
  4. Minoration de la hauteur de Néron-Tate sur les surfaces abéliennes
  5. Stability of flows associated to gradient vector fields and convergence of iterated transport maps
  6. Willmore submanifolds of the Möbius space and a Bernstein-type theorem
  7. A positivity property of the Satake isomorphism
  8. Homological characterization of lean algebras
  9. Reduction of structure for torsors over semilocal rings
  10. Finitistic dimensions of semiprimary rings
  11. Extending Elliptic Curve Chabauty to higher genus curves
  12. The Sp 3 -grassmannian and duality for prime Fano threefolds of genus 9
  13. Existence and compactness for the $${\overline{\partial}}$$ -Neumann operator on q-convex domains
  14. The normalization constant of a certain invariant measure on GL n ()
  15. Tropical descendant Gromov–Witten invariants
  16. A Harnack inequality for the quasi-minima of scalar integral functionals with general growth conditions
  17. Cut loci and conjugate loci on Liouville surfaces
  18. A sphere theorem on locally conformally flat even-dimensional manifolds
  19. On the class numbers of certain Hilbert class fields
  20. The zeros of certain weakly holomorphic Drinfeld modular forms
  21. Icosahedral representations and elliptic curves
  22. On entire functions whose derivatives are¶integer-valued on geometric progressions
  23. Perversité et variation
  24. On linear combinations of special values of L -functions
  25. Irrationalité de certaines sommes de séries
  26. Deformations of Koszul Artin–Schelter Gorenstein algebras
  27. On positive definite hermitian forms

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