2008
DOI: 10.1112/s1461157000000644
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Rational 6-Cycles Under Iteration of Quadratic Polynomials

Abstract: We present a proof, which is conditional on the Birch and Swinnerton-Dyer Conjecture for a specific abelian variety, that there do not exist rational numbers x and c such that x has exact period N = 6 under the iteration x → x 2 + c. This extends earlier results by Morton for N = 4 and by Flynn, Poonen and Schaefer for N = 5.

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Cited by 51 publications
(45 citation statements)
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“…(a) For N ∈ {1, 2, 3}, there are infinitely many c ∈ Q such that f c (x) has a periodic point α ∈ Q of exact period N . (b) (Morton, 1992, [172]; Flynn-Poonen-Schaefer, 1997, [80]; Stoll, 2008, [222])…”
Section: Uniform Boundedness Of (Pre)periodic Pointsmentioning
confidence: 99%
“…(a) For N ∈ {1, 2, 3}, there are infinitely many c ∈ Q such that f c (x) has a periodic point α ∈ Q of exact period N . (b) (Morton, 1992, [172]; Flynn-Poonen-Schaefer, 1997, [80]; Stoll, 2008, [222])…”
Section: Uniform Boundedness Of (Pre)periodic Pointsmentioning
confidence: 99%
“…The curves X 1 (N ), which we define in §2.1, have been studied extensively since the 1980's, beginning with the work of Douady and Hubbard [5] and continuing, for example, in [3,4,20,24]. The main results of [13,25,35] mentioned above involved finding all rational points on the curves X 1 (N ) with N ∈ {4, 5, 6}. However, for providing a classification as in Theorem 1.4 or Conjecture 1.5, one needs a more general notion of a dynamical modular curve which parametrizes quadratic polynomial maps with several marked preperiodic points.…”
Section: Introductionmentioning
confidence: 99%
“…For n ≥ 5, the quotient surface M 2 (n)/ σ n parametrises the set of orbits of size n of endomorphisms of P 1 of degree 2 (see Lemma 2.1). One approach to Poonen's conjecture, carried out in [FPS97,Mor98,Sto08] and elsewhere, is to study the quotient curve C 2 (n)/ σ n , which has a lower genus than C 2 (n).…”
Section: Introductionmentioning
confidence: 99%