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